Two ^9Li clusters connected with two valence neutrons in ^{20}C
aa r X i v : . [ nu c l - t h ] F e b Two Li clusters connected with two valence neutrons in C Naoyuki Itagaki, Tokuro Fukui, Junki Tanaka, and Yuma Kikuchi Yukawa Institute for Theoretical Physics, Kyoto University,Kitashirakawa Oiwake-Cho, Kyoto 606-8502, Japan RIKEN Nishina Center for Aaccelerator-Based Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Tokuyama College of Technology, Gakuendai, Shunan, Yamaguchi 745-8585, Japan (Dated: February 10, 2021)Many preceding works have shown in Li the presence of the halo structure comprised of theweakly bound two neutrons around Li, and it is intriguing to see how this halo structure changeswhen another Li approaches. In this study, we introduce a four-body model for C with two Li clusters and two valence neutrons. The recent development of the antisymmetrized quasi clus-ter model (AQCM) makes it possible to generate jj -coupling shell-model wave functions from α cluster models. Here, jj -coupling shell model wave function of Li is regarded as a cluster, whichcorresponds to the subclosure configuration of p / for the neutrons, and we discuss how the twoneutrons connect two Li clusters. Until now, most of the clusters in the conventional models havebeen limited to the closures of the three-dimensional harmonic oscillators, such as He, O, and Ca; however, owing to AQCM, it is feasible to utilize the jj -coupling shell model wave functionsas plural subsystems quite easily. The appearance of a rotational band structure with a clusterstructure around the four-body threshold energy is discussed. I. INTRODUCTION
The He nuclei have been known to have large bindingenergy (28.3 MeV) distinctly in the light mass region.In contrast, it is known that the relative interaction be-tween He nuclei is weak. Therefore, they can serve assubsystems in some of the light nuclei, called α clusterstructure [1, 2]. The α cluster structure has been studiedfor decades, and one of the most famous examples is thesecond 0 + state of C with a developed three- α clusterstructure [3, 4], which is called the Hoyle state. Manycluster models have proven to be capable of describingvarious properties of the Hoyle state [5, 6].In most of the conventional cluster models, however,only the states corresponding to the closure of the three-dimensional harmonic oscillator, such as He, O, and Ca, have been treated as subsystems called clusters. Inthese cases, unfortunately, the contribution of the non-central interactions (spin-orbit and tensor interactions),which are quite important in the nuclear systems, van-ishes after the antisymmetrization of the wave functions.This is the consequence of the fact that the closure con-figurations of the major shells are spin-zero systems, andthe non-central interactions do not contribute to suchspinless systems. In actual nuclear systems, on the con-trary, the contribution of the spin-orbit interaction isessential, which enables to explain the observed magicnumbers; the subclosure configurations of the jj -couplingshell model ( f / , g / , and h / ) correspond to the ob-served magic numbers of 28, 50, and 126 [7]. If we enlargethe model space of the cluster models and open the pathto another symmetry, indeed this spin-orbit interactionworks as a driving force to break the α clusters, for in-stance in C [8].Therefore, the important task is to include the spin-orbit contribution by extending the traditional clustermodels; we proposed the antisymmetrized quasi cluster model (AQCM) [9–21]. This method allows the smoothtransformation of the α cluster model wave functions to jj -coupling shell model ones. We call the clusters thatfeel the spin-orbit contribution after this transformationquasi clusters. The conventional α cluster models coverthe model space of closure of major shells (correspond-ing to the magic numbers of 2, 8, and 20), and now thesubclosure configurations of the jj -coupling shell model, p / , d / , f / , g / etc. , are covered by our AQCM [15].The achievement of AQCM allows us to use jj -couplingshell model wave functions as subsystems of the nu-clei, which is the beginning of a new cluster model. Atfirst, we have shown the possibility of C, He, and Li as clusters [20, 21], where subclosure configurationof p / plays an important role. The cluster structuresof Be ( He+ He), B ( He+ Li), C ( Li+ Li), C(three He), and Ar (three C), have been investigatedaround the corresponding threshold energies.In this study, we further add neutrons and discuss in C the Li+ Li cluster configuration as an example.It has been known in Li that the two neutrons haveneutron halo structure around Li [22]. It is intriguingto see how such structure is affected when another Liapproaches to the halo neutrons. In the Be isotopes,it has been extensively discussed that neutrons performmolecular-orbital motion around α clusters; very devel-oped α - α cluster structure appears when two neutronsoccupy the σ orbit [23, 24]. This idea can be extendedto the linear-chain structure of three α clusters in the Cisotopes [25, 26]. If the neutrons in the halo state per-form molecular orbital motion around two Li clusterswith large distances, it paves the way to a novel bind-ing mechanism in the excited states of the neutron-richnuclei.This paper is organized as follows. The framework isdescribed in Sec. II. The results are shown in Sec. III.The conclusions are presented in Sec. IV.
II. FRAMEWORKA. Basic feature of AQCM
AQCM allows the smooth transformation of the clustermodel wave functions to the jj -coupling shell model ones.In AQCM, each single particle is described by a Gaussianform as in many other cluster models including the Brinkmodel [1], φ τ,σ ( r ) = (cid:18) νπ (cid:19) exp h − ν ( r − ζ ) i χ τ,σ , (1)where the Gaussian center parameter ζ is related to theexpectation value of the position of the nucleon, and χ τ,σ is the spin-isospin part of the wave function. For the sizeparameter ν , here we use ν = 0 .
20 fm − , which gives theoptimal 0 + energy of C within a single AQCM basisstate. The Slater determinant is constructed from thesesingle-particle wave functions by antisymmetrizing them.Next, we focus on the Gaussian center parameters { ζ i } .As in other cluster models, here four single-particle wavefunctions with different spin and isospin sharing a com-mon ζ value correspond to an α cluster. This clusterwave function is transformed into jj -coupling shell modelbased on the AQCM. When the original value of theGaussian center parameter ζ is R , which is real and re-lated to the spatial position of this nucleon, it is trans-formed by adding the imaginary part as ζ = R + i Λ e spin × R , (2)where e spin is a unit vector for the intrinsic-spin orien-tation of this nucleon. The control parameter Λ is as-sociated with the breaking of the cluster, and with afinite value of Λ, the two nucleons with opposite spinorientations have the ζ values, which are complex conju-gate with each other. This situation corresponds to thetime-reversal motion of two nucleons. After this trans-formation, the α clusters are called quasi clusters. Wecan generally create the single-particle orbits of the jj -coupling shell model by taking the limits of R → → B. Wave function for C The total wave function for C is the superposition ofdifferent Slater determinants, { Φ i } ,Ψ J π = X i,K c Ki P KJ π Φ i . (3)All Slater determinants are projected to the eigen statesof parity and angular momentum by using the projectionoperator P KJ π , P KJ π = P π J + 18 π Z d Ω D JMK ∗ R (Ω) . (4) Here D JMK is the Wigner D -function and R (Ω) is therotation operator for the spatial and spin parts of thewave function. This integration over the Euler angle Ω isnumerically performed. The operator P π is for the parityprojection ( P π = (1 + P r ) / √ P r is the parity-inversion operator), whichis also performed numerically. The coefficients (cid:8) c Ki (cid:9) areobtained together with the energy eigenvalue E whenwe diagonalize the norm and Hamiltonian ( H ) matrices,namely by solving the Hill-Wheeler equation. X i,j,K,K ′ ( < Φ i | ( P K ′ J π ) † HP KJ π | Φ j > − E < Φ i | ( P K ′ J π ) † P KJ π | Φ j > ) c Kj = 0 . (5)This angular momentum projection enables to generatedifferent K number states as independent basis statesfrom each Slater determinant.Each Slater determinant consists of the antisym-metrized product of single-particle wave functions.Φ i = A{ φ τ ,σ ( r , ζ ) φ τ ,σ ( r , ζ ) · · ·· · · φ τ ,σ ( r , ζ ) φ τ ,σ ( r , ζ ) } i . (6)Here, the single-particle wave functions from φ τ ,σ ( r , ζ ) to φ τ ,σ ( r , ζ ) belong to one Li cluster,whereas those from φ τ ,σ ( r , ζ ) to φ τ ,σ ( r , ζ )are for another Li. For each Li cluster, we introducethe subclosure configuration of ( s / ) ( p / ) for theneutrons. For the proton part, the last protons in two Liclusters are introduced as time-reversal configurations, | / / i and | / − / i . Then these two Li clustersare separated with the relative distance of d i , whichis randomly generated with equal probability between0.5 fm and 5.0 fm.The two valence neutrons, φ τ ,σ ( r , ζ ) and φ τ ,σ ( r , ζ ), are introduced to have opposite spindirections (spin-up and spin-down). Their Gaussian cen-ter parameters ( ζ and ζ ) are generated by using ran-dom numbers { r k } , which have the probability distribu-tion P ( | r k | ) proportional to exp [ − | r k | /σ ], P ( | r k | ) ∝ exp [ − | r k | /η ] . (7)The value of η is chosen to be 1 . { r k } ,we multiply the sign factor to each r k , which allows r k to be positive and negative with equal probability. Theresultant random numbers are applied to all three ( x , y , z ) components of the Gaussian center parameters of thetwo valence neutrons. C. Hamiltonian
The Hamiltonian consists of the kinetic energy andpotential energy terms. For the potential part, theinteraction consists of the central ( ˆ V central ), spin-orbit( ˆ V spin-orbit ), and Coulomb terms. For the central part,the Tohsaki interaction [27] is adopted. This interactionhas finite ranges for the three-body terms in additionto two-body terms, which is designed to reproduce bothsaturation properties and scattering phase shifts of two α clusters. For the spin-orbit part, we use the spin-orbitterm of the G3RS interaction [28], which is a realistic in-teraction originally developed to reproduce the nucleon-nucleon scattering phase shifts.The Tohsaki interaction consists of two-body ( V (2) )and three-body ( V (3) ) terms:ˆ V central = 12 X i = j V (2) ij + 16 X i = j,j = k,i = k V (3) ijk , (8)where V (2) ij and V (3) ijk have three ranges, V (2) ij = X α =1 V (2) α exp " − ( ~r i − ~r j ) µ α W (2) α − M (2) α P σ P τ (cid:17) ij , (9) V (3) ijk = X α =1 V (3) α exp " − ( ~r i − ~r j ) µ α − ( ~r i − ~r k ) µ α × (cid:16) W (3) α − M (3) α P σ P τ (cid:17) ij (cid:16) W (3) α − M (3) α P σ P τ (cid:17) ik . (10)Here, P σ P τ represents the exchange of the spin-isospinpart of the wave functions of interacting two nucleons.The physical coordinate for the i th nucleon is ~r i . Thedetails of the parameters are shown in Ref. [27], but weuse F1’ parameter set for the Majorana parameter ( M (3) α )of the three-body part introduced in Ref. [9].The G3RS interaction [28] is a realistic interaction, andthe spin-orbit term has the following form;ˆ V spin-orbit = 12 X i = j V lsij , (11)where V lsij = (cid:16) V ls e − d ( ~r i − ~r j ) − V ls e − d ( ~r i − ~r j ) (cid:17) P (cid:0) O (cid:1) ~L · ~S. (12)Here, ~L is the angular momentum for the relative mo-tion between the i th and j th nucleons, and ~S is the sumof the spin operator for these two interacting nucleons.The operator P (cid:0) O (cid:1) stands for the projection onto thetriplet-odd state. The strength of the spin-orbit inter-actions is set to V ls = V ls = 1800 MeV, which allowsconsistent description of C and O [9].
III. RESULTSA. Energy convergence of C We start the discussion with the 0 + energy curves of C calculated with the present Li+ Li+ n + n model.
10 20−30−20−100 C number of the basis states E ne r g y o f t he + s t a t e s ( M e V ) FIG. 1. Energy curves for the 0 + states of C calculatedwith the Li+ Li+ n + n model. Horizontal axis shows thenumber of Slater determinants superposed. Energies are mea-sured from the four-body threshold. The horizontal axis of Fig. 1 shows the number of Slaterdeterminants superposed, and for each Slater determi-nant, the distances between two Li ( d ) and the positionsfor the Gaussian center parameters for the two valenceneutrons ( ζ and ζ ) are randomly generated. The en-ergies are measured from the four-body threshold. Thiscalculation is based on the bound state approximationand it could be possible that some of the obtained statesabove the neutron threshold are continuum states, butthe states with flat energies after rapid convergence arecandidates for the resonance states. The lowest 0 + stateconverges to − . C is located at − . + energies are summarized in Table I. B. Principal quantum number of the harmonicoscillator
We discuss the property of each state obtained after su-perposing the basis states. Particular focus is placed onthe point which states have the character of Li+ Li clus-ter configuration. One of the physical quantities, whichcharacterize each state, is the expectation value of theprincipal quantum number ˆ N of the harmonic oscillator,ˆ N = X i a † i · a i . (13)The summation can be taken independently for the pro-ton part and neutron part. In Table I, the column “ N -proton” stands for the expectation value of ˆ N for theprotons and “ N -neutron” for the neutrons. The lowestPauli-allowed values are 4 for the protons (two are in the s -shell and four are in the p -shell) and 18 for the neutrons(two are in the s -shell, six are in the p -shell, and six are inthe sd -shell). The lowest 0 + state has the values of 4.19(protons) and 18.61 (neutrons) fairly close to the lowestPauli-allowed values, which indicates that the state has TABLE I. Properties of 0 + states of C obtained by su-perposing Slater determinants. The energies are measuredfrom the Li + Li + n + n threshold ( E (MeV)). Expecta-tion values of the principal quantum number for the protonpart ( N -proton) and neutron part ( N -neutron) are listed to-gether with those of the single-particle parity of the protons( spp -proton) and neutrons ( spp -neutron). E (MeV) N -proton N -neutron spp -proton spp -neutron1 − .
37 4 .
19 18 . − .
90 1 . − .
56 4 .
43 18 . − .
78 1 . − .
34 4 .
90 19 . − .
56 1 . − .
02 4 .
83 19 . − .
58 1 . − .
75 5 .
60 21 . − . − . − .
03 4 .
71 19 . − .
59 1 . − .
03 5 .
68 21 . − .
29 1 . − .
96 6 .
12 21 . − .
35 0 . − .
02 5 .
77 21 . − .
13 1 . .
38 5 .
96 22 . − .
09 0 . pure shell-model character. However, the excited stateshave much larger values.It should be stressed that the principal quantum num-ber N for the protons ( N -proton) is governed by thedistance between two Li clusters, since protons are onlyin the Li clusters and the internal wave functions ofeach Li is frozen except for the antisymmetrization ef-fect. Thus, the N -proton values listed in Table I is con-sidered to contain information for the distance betweentwo Li clusters, although the expectation values are ob-tained after superposing Slater determinants with differ-ent Li- Li distances. To extract the contribution of the Li + Li core part, in Fig. 2, we show the N values of C without the two valence neutrons as a function ofthe distance between two Li clusters (Fig. 2(a): protonsand Fig. 2(b): neutrons). At the zero-limit for the dis-tance d between two Li, the values are 4 (protons) and14 (neutrons), The values increase to 4.9 (protons) and15.7 (neutrons) at d = 3 fm, and they grow to 5.6 and16.9 at d = 3 . d = 4 fm).According to Table I, the fifth state at − .
75 MeV,seventh state at − .
03 MeV, eighth state at − .
96 MeV,ninth state at − .
02 MeV, and tenth state at 1 .
38 MeVhave the N -proton values of 5 . − .
0. Thus, it is quitelikely that these states have the cluster structure withthe Li- Li distances of 3 . − . − .
96 MeV has the largest N -protonvalue of 6.12, and we consider this state as a candidatefor the cluster state representing these states. The N -proton value of 6.12 corresponds to the Li- Li distancesof 3 .
75 fm in C without the valence neutrons (Fig. 2(a)).In our preceding work for C [21], Li + Li cluster statehas been shown to appear slightly below the correspond-ing threshold energy. Therefore, it is rather reasonable to Li− Li distance (fm) C protons P r i n c i pa l quan t u m nu m be r (a)0 2 4101520 Li− Li distance (fm) C neutrons P r i n c i pa l quan t u m nu m be r (b) FIG. 2. Principal quantum number N of the harmonic os-cillator for the 0 + state of C as a function of the distancebetween two Li clusters. (a): protons and (b): neutrons. find here the appearance of the cluster state in C afteradding two valence neutrons to C below the four-bodythreshold. This state shows rapid energy convergencein Fig 1, which is considered to be a possible resonancestate.If we assume that the 0 + state of C at − .
96 MeV,which is the candidates for the cluster state, has the Li- Li distances of about 3.75 fm, the neutrons in the Li+ Li core part must have the N -neutron value around17.75 (Fig. 2(b)). Therefore, we can deduce the con-tribution of the two valence neutrons in the state at − .
96 MeV by subtracting 17.75 from the N -neutronvalue of 21.79 listed in Table I. The two valence neu-trons are estimated to have the N value of about 4.0 inthe cluster state, and each valence neutron shares thevalue of ∼ C. Single-particle parity
To discuss the properties of the two valence neutrons in C in more detail, next we calculate the single-particleparity. In this framework, the single-particle orbits in-troduced are non-orthogonal, and therefore we cannotdirectly discuss the parity of each orbit. Another dif-ficulty of directly discussing the parity of each single-particle comes from the superposition of Slater determi-nants performed here. Nevertheless, we can get insightsfor the single-particle parity from the expectation valueof the single-particle operator, which is the sum of theparity inversion operator for each nucleon [29],ˆ O spp = X i P ri . (14)Here P ri is the parity-inversion operator for the i -th nu-cleon. The eigenvalues of P ri is 1 and −
1, for the positive-parity orbit and negative-parity orbit, respectively. Thesummation can be taken over the protons or neutronsindependently.We calculate the expectation value of this single-particle parity for the 0 + states of C listed in Table I,which are obtained by superposing Slates determinants.In Table I, the column “ spp -proton” is for the protonsand “ spp -neutron” is for the neutrons. The ground statehas the values of − .
90 (protons) and 1.85 (neutrons)indicating that the state has the character of the lowestshell-model state. In the lowest shell-model state, twoprotons and two neutrons are in the s -shell with positive-parity, four protons and six neutrons are in the p -shellwith negative-parity, and six neutrons are in the sd -shellwith positive-parity, and hence, the single-particle paritybecomes 2 − − − − .
96 MeV in Table I identified ascluster state in the previous subsection) has quite differ-ent values; − .
09 ( spp -proton) and 0.08 ( spp -neutron).In the present model, the protons are only in the Li + Li core part as mentioned before. Thus, the calcu-lated single-particle parity of the protons ( spp -proton)is governed by the distance between two Li clusters.The correspondence between the distance between two Li clusters and spp -proton can be clarified in C by re-moving the two valence neutrons. Figure 3 depicts thesingle-particle parity for the 0 + state of C as a functionof the distance between two Li clusters, where the dottedline is for the protons ( spp -proton) and the dashed-line isfor the neutrons ( spp -neutron). For the protons (dottedline), the value is around − Li occupies p -shell-likeone-node orbit. However the value starts slightly devi-ating from − Li sys-tem is always projected to positive. The value for theprotons is − .
60 at the d = 3 . − . d = 4 fm.We can compare this result with the spp -proton valuesof C listed in Table I, where the two valence neutronsare added and the states with different Li- Li distancesare superposed. In the previous subsection, we have iden-tified that the eighth state in Table I as the candidates forthe cluster state with the Li- Li distances of ∼ .
75 fm. Li− Li distance (fm) Cprotons S i ng l e pa r t i c l d pa r i t y neutrons FIG. 3. Single-particle parity for the 0 + state of C as afunction of the distance between two Li clusters. Dotted lineis for the protons and dashed-line is for the neutrons.
The obtained spp -proton values of C is − .
35 for theeighth state, quite different from −
2. The obtained valueis consistent with that for the states having finite Li- Lidistances, suggesting the prominent cluster structure.Next, we discuss the single-particle parity of the neu-tron part. It is intriguing to point out that in C with-out the two valence neutrons, the single-particle parityof the neutrons is always zero independent of the Li- Lidistance (Fig. 3 dashed line). This is coming from thefact that the neutrons in both Li have identical (sub-closure) configurations except for the central positions ofthe clusters. The antisymmetrization effect allows thelinear combination of the single-particle orbits, and thecombinations of the two orbits with good parity aroundthe center of left or right cluster always create a pairof positive-parity and negative-parity orbits. Thus, thesingle-particle parity for the neutrons of C is zero, andin C, the spp -neutron values purely show the contribu-tion of the two valence neutrons. In Table I, the groundstate and low excited states have the value close to 2indicating that the two valence neutrons are in the sd -shell-like positive-parity orbits. However, the candidatefor the cluster state (eighth state in Table I) have thevalue of 0.08 completely different from 2. The resultindicates that the valence neutrons have more complexcharacters than “pure two-node orbits”, which was notevident when we discussed the principal quantum num-ber N in the previous subsection. One of the possibleexplanations is that the orbits of the valence neutronshave mixed components of the p -shell, sd -shell, and pf -shell orbits. D. Rotational band structure of C The same calculation can be performed also for the 2 + and 4 + states of C. The eighth states in Table I at − .
96 MeV, which is a candidate for the cluster struc-ture, serves as a band head state of a rotational band −20−10010 0 + + +20 C FIG. 4. Rotational band structure of C measured from the Li + Li + n + n threshold energy. Solid, dotted, and dashedlines are for 0 + , 2 + , and 4 + states. structure as shown in Fig. 4, where the solid, dotted, anddashed lines are for 0 + , 2 + , and 4 + states, respectively.The 2 + state at − .
29 MeV and 4 + states at 0 .
33 MeVare connected with the 0 + state at − .
96 MeV and clas-sified into a rotational band judging from the similarityof the calculated properties (principal quantum numberand single-particle parity) and electromagnetic transitionprobabilities among the states. We have not consideredthe energy of each state in this classification, but eventu-ally, typical level spacings of the rotational band struc-ture comes out around the Li + Li + n + n thresholdenergy. The rotational band shows small level spacingsbetween 0 + and 2 + reflecting the large deformation. IV. CONCLUSIONS
In this study, we discussed the structure of C by in-troducing a four-body cluster model with Li+ Li+ n + n configurations. The recent development of the antisym- metrized quasi cluster model (AQCM) enables us to uti-lize jj -coupling shell-model wave functions as plural sub-systems quite easily. Until now, many works have shownin Li the presence of the halo structure comprised ofthe weakly bound two neutrons around Li, and ourstudy was motivated by a question how this halo struc-ture changes when another Li approaches.In our preceding work for C [21], it has been dis-cussed that Li + Li cluster state appears below thethreshold, and therefore, it is natural in C to see theappearance of the cluster states below the threshold afteradding the two valence neutrons. The candidate for thecluster state was identified by calculating the principalquantum number and single-particle parity. The candi-date for the cluster state serves as a band head state andforms a rotational band structure, where the members arecollected by the similarity of the properties and electro-magnetic transition probabilities among the states. Thestates have small level spacings reflecting the large defor-mation.As future work, we investigate the possibility ofmolecular-orbital states. In the present analysis, we dis-cussed that atomic-orbit states, where neutron(s) sticksto one of the two centers, but in the molecular-orbitstate, each valence neutron rotates around two centersequally with good parity. Such molecular-obit states areexpected to appear around the threshold energy. Also,we will connect the calculation to the nuclear reactionand discuss how we can populate these states in the ac-tual experiment.
ACKNOWLEDGMENTS
The authors would like to thank the discussion withDr. M. Sasano (RIKEN). The numerical calculationshave been performed using the computer facility ofYukawa Institute for Theoretical Physics, Kyoto Univer-sity. [1] D. M. Brink, Proc. Int. School Phys.“Enrico Fermi”
XXXVI , 247 (1966).[2] M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, andU.-G. Meißner, Rev. Mod. Phys. , 035004 (2018).[3] F. Hoyle, Astrophys. J. Suppl. , 121 (1954).[4] M. Freer and H. Fynbo,Prog. Part. Nucl. Phys. , 1 (2014).[5] Y. Fujiwara, H. Horiuchi, K. Ikeda, M. Kamimura,K. Kat¯o, Y. Suzuki, and E. Uegaki, Prog. Theor. Phys.Supplement , 29 (1980).[6] A. Tohsaki, H. Horiuchi, P. Schuck, and G. R¨opke,Phys. Rev. Lett. , 192501 (2001).[7] M. G. Mayer and H. G. Jensen, “Elementary theory ofnuclear shell structure”, John Wiley, Sons, New York,Chapman, Hall, London (1955).[8] N. Itagaki, S. Aoyama, S. Okabe, and K. Ikeda,Phys. Rev. C , 054307 (2004). [9] N. Itagaki, Phys. Rev. C , 064324 (2016).[10] N. Itagaki, H. Masui, M. Ito, S. Aoyama, and K. Ikeda,Phys. Rev. C , 034310 (2006).[11] H. Masui and N. Itagaki,Phys. Rev. C , 054309 (2007).[12] T. Yoshida, N. Itagaki, and T. Otsuka,Phys. Rev. C , 034308 (2009).[13] N. Itagaki, J. Cseh, and M. P loszajczak,Phys. Rev. C , 014302 (2011).[14] T. Suhara, N. Itagaki, J. Cseh, and M. P loszajczak,Phys. Rev. C , 054334 (2013).[15] N. Itagaki, H. Matsuno, and T. Suhara, Prog. Theor.Exp. Phys. , 093D01 (2016).[16] H. Matsuno, N. Itagaki, T. Ichikawa, Y. Yoshida,and Y. Kanada-En’yo, Prog. Theor. Exp. Phys. ,063D01 (2017).[17] H. Matsuno and N. Itagaki, Prog. Theor. Exp. Phys. , 123D05 (2017).[18] N. Itagaki and A. Tohsaki,Phys. Rev. C , 014307 (2018).[19] N. Itagaki, H. Matsuno, and A. Tohsaki,Phys. Rev. C , 044306 (2018).[20] N. Itagaki, A. V. Afanasjev, and D. Ray,Phys. Rev. C , 034304 (2020).[21] N. Itagaki, T. Fukui, J. Tanaka, and Y. Kikuchi,Phys. Rev. C , 024332 (2020).[22] I. Tanihata, H. Hamagaki, O. Hashimoto,Y. Shida, N. Yoshikawa, K. Sugimoto, O. Ya-makawa, T. Kobayashi, and N. Takahashi, Phys. Rev. Lett. , 2676 (1985).[23] N. Itagaki and S. Okabe,Phys. Rev. C , 044306 (2000).[24] M. Ito, N. Itagaki, H. Sakurai, and K. Ikeda,Phys. Rev. Lett. , 182502 (2008).[25] N. Itagaki, S. Okabe, K. Ikeda, and I. Tanihata,Phys. Rev. C , 014301 (2001).[26] P. W. Zhao, N. Itagaki, and J. Meng,Phys. Rev. Lett. , 022501 (2015).[27] A. Tohsaki, Phys. Rev. C , 1814 (1994).[28] R. Tamagaki, Prog. Theor. Phys. , 91 (1968).[29] N. Itagaki, W. von Oertzen, and S. Okabe,Phys. Rev. C74