aa r X i v : . [ nu c l - t h ] M a y Structure and decay pattern of linear-chain state in C T. Baba and M. Kimura
Department of Physics, Hokkaido University, 060-0810 Sapporo, Japan (Dated: October 6, 2018)The linear-chain states of C are theoretically investigated by using the antisymmetrized molec-ular dynamics. The calculated excitation energies and the α decay widths of the linear-chain stateswere compared with the observed data reported by the recent experiments. The properties of thepositive-parity linear-chain states reasonably agree with the observation, that convinces us of thelinear-chain formation in the positive-parity states. On the other hand, in the negative-parity states,it is found that the linear-chain configuration is fragmented into many states and do not form asingle rotational band. As a further evidence of the linear-chain formation, we focus on the α decaypattern. It is shown that the linear-chain states decay to the excited states of daughter nucleus Beas well as to the ground state, while other cluster states dominantly decay into the ground state.Hence, we regard that this characteristic decay pattern is a strong signature of the linear-chainformation.
I. INTRODUCTION
A variety of the α cluster structure are known to ex-ist in light stable nuclei. The most famous example isthe Hoyle state (the 0 +2 state of C) whose dilute gas-like 3 α cluster structure has been studied in detail [1–8] and identified well today. The linear-chain configu-ration of 3 α clusters in which α particles are linearlyaligned is an other example of famous and exotic clus-ter structure. It was firstly suggested by Morinaga [9]to explain the structure of the Hoyle state. However, asmentioned above, it turned out that the Hoyle state doesnot have the linear-chain configuration but has the di-lute gas-like nature. In addition to this, the instabilityof the linear-chain configuration against the bending mo-tion (deviation from the linear alignment) was pointedout by the antisymmetrized molecular dynamics (AMD)[4] and Fermionic molecular dynamics (FMD) calcula-tions [7]. Thus, the formation of perfectly linear-alignedconfiguration in C ∗ looks negative despite of the manyyears of research.The interest in the linear-chain configuration is rein-forced by the unstable nuclear physics, because the va-lence neutrons may stabilize it by their glue-like role.Such glue-like role of valence neutron is well known forBe isotopes in which 2 α cluster core is assisted by thevalence neutrons occupying the molecular-orbits [10–15].As a natural consequence, we expect that the linear-chainconfiguration can be realized in neutron-rich C isotopes,and this expectation have been motivating many theoret-ical and experimental studies [16–28]. Recently, ratherpromising candidates of linear-chain configuration in Cwere independently reported by two groups [29, 30]. Bothgroups observed the He + Be resonances above the α threshold energy in both of positive- and negative-parity. The reported energies of the positive-parity reso-nances measured from the α threshold are in reasonableagreement with the excitation energies of the linear-chainstates predicted by Suhara et al. [24] on the basis ofthe antisymmetrized molecular dynamics (AMD) calcula-tion. Thus, rather promising evidence of the linear-chain formation has been found.However, there are still several gaps between theoryand experiment which must be resolved to confirm thelinear-chain formation in C. First, when measured fromthe ground state energy, theoretically predicted and ex-perimentally observed excitation energies of the positive-parity resonances disagree. This may be because theeffective interaction used in the calculation [31] do notreproduce the α threshold energy. Second, the experi-ments report the negative-parity resonances, while thenegative-parity linear-chain states were not clearly iden-tified in Ref. [24]. Finally, the experiment [29] reportedthe α decay width of the resonances which is a strongevidence of the α clustering and must be verified by thetheoretical calculation. Thus, further theoretical studiesare in need to identify the linear-chain states in C.For this purpose, we investigated the linear-chainstates in C. For the sake of the quantitative compar-ison of the excitation energy, we performed AMD cal-culation employing Gogny D1S effective interaction [32]which reproduces threshold energies in C. From theAMD wave function, we estimated the α decay widthsof the linear-chain states as well as those of other clusterand non-cluster states. It is found that the calculated ex-citation energies of the positive-parity linear-chain statesare in good agreement with the observation, and onlythe linear-chain states have large α decay widths compa-rable with the observed data. Hence, we consider thatthe linear-chain formation in the positive-parity state israther plausible. On the other hand, in the negative-parity state, the linear-chain configuration is fragmentedinto many states and do not form a single rotationalband. As a further evidence of the linear-chain forma-tion, we focus on the α decay pattern. It is shown thatthe linear-chain states decay to the excited states of Beas well as the Be ground state, while other cluster statesdominantly decay to the Be ground state. This char-acteristic decay pattern is, if it is observed, a strong sig-nature of the linear-chain formation.The contents of this paper are as follows. In the nextsection, the AMD framework and the method to esti-mate the alpha decay width are briefly outlined. In theIII section, the results of the energy variation and gener-ator coordinate method are presented. In the section IV,the energies and α width of the linear-chain states arecompared with the observed data. In the last section, wesummarize this study. II. THEORETICAL FRAMEWORKA. variational calculation and generator coordinatemethod
The microscopic A -body Hamiltonian used in thisstudy reads,ˆ H = A X i =1 ˆ t i + A X i Nucleon single-particle energy and orbit are useful toinvestigate the motion of the valence neutrons aroundthe core nucleus. For this purpose, we construct single-particle Hamiltonian and calculate the neutron single-particle orbits in the intrinsic wave function Φ int ( β, γ ).We first transform the single particle wave packet ϕ i tothe orthonormalized basis, e ϕ α = 1 √ λ α A X i =1 g iα ϕ i . (12)Here, λ α and g iα are the eigenvalues and eigenvectors ofthe overlap matrix B ij = h ϕ i | ϕ j i . Using this basis, thesingle particle Hamiltonian is derived, h αβ = h e ϕ α | ˆ t | e ϕ β i + A X γ =1 h e ϕ α e ϕ γ | ˆ v N + ˆ v C | e ϕ β e ϕ γ − e ϕ γ e ϕ β i , + 12 A X γ,δ =1 h e ϕ γ e ϕ δ | e ϕ ∗ α e ϕ β δ ˆ v N δρ | e ϕ γ e ϕ δ − e ϕ δ e ϕ γ i . (13)The eigenvalues ǫ s and eigenvectors f αs of h αβ give thesingle particle energies and the single particle orbits, e φ s = P Aα =1 f αs e ϕ α . We also calculate the amount of thepositive-parity component in the single-particle orbit, p + = |h e φ s | P x | e φ s i| , (14)and angular momenta in the intrinsic frame, j ( j + 1) = h e φ s | ˆ j | e φ s i , | j z | = q h e φ s | ˆ j z | e φ s i , (15) l ( l + 1) = h e φ s | ˆ l | e φ s i , | l z | = q h e φ s | ˆ l z | e φ s i , (16)which are used to discuss the properties of the singleparticle orbits. C. α reduced width amplitude and decay width From the GCM wave function, we estimate the α re-duced width amplitude (RWA) y lj π ′ ( r ) which is definedas y lj π ′ ( r ) = s A !4!( A − h φ α [ φ Be ( j π ′ ) Y l (ˆ r )] J π M | Ψ J π Mn i , (17)where φ α and φ Be ( j π ′ ) denote the wave functions for α particle and daughter nucleus Be with spin-parity j π ′ .The square of the RWA | ry lj π ′ ( r ) | is equal to the prob-ability to observe α particle and Be with spin-parity j π ′ at inter-cluster distance r with relative orbital an-gular momentum l in the C described by GCM wavefunction Ψ J π Mn .Using RWA, the partial α decay widths for the decayprocess C( J π ) → α + Be( j π ′ ) is estimated asΓ αlj π ′ = 2 P l ( a ) γ lj π ′ ( a ) , P l ( a ) = kaF l ( ka ) + G l ( ka ) , (18) where a denote the channel radius, and the penetrationfactor P l is given by the Coulomb regular and irregularwave functions F l and G l . The wave number k is given bythe decay Q -value and the reduced mass as k = p µE Q .The reduced width γ lj π ′ is γ lj π ′ ( a ) = ~ µa [ ay lj π ′ ( a )] . (19)To calculate RWA with reduced computational cost,we employ the method given in Ref. [39] which suggestsan approximation validated for sufficiently large inter-cluster distance a , | ay lj π ′ ( a ) | ≃ r γ π | h Φ JπBB ( a ) | Ψ J π Mn i | , (20) γ = 4( A − A ν BB , which means that RWA is reasonably approximated bythe overlap between the GCM wave function and theBrink-Bloch wave function Φ J π BB ( a ) composed of the Beand α particle with the Gaussian width parameter ν BB .In the case of the present study, since Be is deformed,it must be projected to the eigenstate of the angular mo-mentum. Therefore, we constructed Φ J π BB as follows. Wefirst calculate the intrinsic wave function ψ Be for Be bythe AMD energy variation in which the width parameter ν BB is fixed to 0.16 fm − . We obtained two different con-figurations for Be so-called π and σ configurations. It isknown that the former is dominant in the ground bandand the latter is dominant in the excited band. Thenthe Be wave function is projected to the eigenstate ofthe angular momentum, and combined with the α clusterwave function to constitute the intrinsic wave function of C ∗ | Φ BB ( a ) i = (cid:12)(cid:12)(cid:12)(cid:12) A (cid:26) ψ α (cid:18) − a (cid:19) ˆ P jm ψ Be (cid:18) a (cid:19) (cid:27)(cid:29) , (21)where we assumed that ψ Be is parity and axially sym-metric, that is validated by the numerical check. Then,the reference wave function is constructed by the angularmomentum projection of the total system.Φ JπBB ( a ) = 1 N l + 12 J + 1 X m C Jml jm ˆ P J π Mm | Φ BB ( a ) i , (22)Here C Jml jm and N denotes the Clebsch-Gordan coeffi-cient and the normalization factor. The summation over m is needed to project the relative angular momentumbetween Be and α particle to l . III. RESULTSA. Energy surface and intrinsic structures Figure 1 (a) shows the energy surface as functions ofquadrupole deformation parameters β and γ for J π = 0 + β γ [degree] γ [degree] β [MeV] [MeV] (a) (b) FIG. 1. (color online) The angular momentum projected en-ergy surface for (a) the J π = 0 + state and (b) J π = 1 − stateas functions of quadrupole deformation parameters β and γ .The circles show the position of the energy minima. (a) (b) (c) (d)(e) (f) (g) (h) 00 -6-6 -4-4 -2-2 22 44 660-6-4-2246 0-6 -4 -2 2 4 6 0-6 -4 -2 2 4 6 0-6 -4 -2 2 4 6 [fm] [f m ] FIG. 2. (color online) The density distribution of (a)-(d) thepositive states and (e)-(h) negative parity states. The con-tour lines show the proton density distributions. The colorplots show the single particle orbits occupied by the mostweakly bound neutron. Open boxes show the centroids of theGaussian wave packets describing protons. states obtained by the constraint variational calculationand angular momentum projection. The circles on theenergy surfaces show the position of the energy minima.The energy minimum of the 0 + state is located at( β, γ ) = (0 . , ◦ ) with the binding energy of 106.1MeV. It is interesting that this minimum state is de-formed, as seen in its intrinsic density distribution shownin Fig. 2 (a), despite of the neutron magic number N = 8.However, the deformation is not large enough to breakthe neutron magicity as the last valence neutron occupies p -wave that can be deduced from the density distributionof the valence neutron orbit shown in Fig. 2 (a).In the oblate deformed region, the different structurewhich we call triangular configuration appears. Fig. 2(b) shows the density distribution of the wave functionlocated at ( β, γ ) = (0 . , ◦ ). The proton density distri-bution have triangular shape showing possible formationof 3 α cluster core with triangle configuration. Indeed, asmentioned later, the excited states composed of this con-figuration have larger α reduced widths than the groundstate. Owing to the parity asymmetric shape, the va-lence proton orbit is an admixture of the positive- and TABLE I. The properties of the most weakly bound protonand neutron orbits in the configurations shown in Fig. 2 (a)-(h). The column occ. shows the number of the nucleon oc-cupying the orbit. When two valence nucleons occupy thealmost degenerated orbits, the single-particle properties areaveraged and occ. is equal to 2. Other columns show thesingle particle energy ε in MeV, the amount of the positive-parity component p + and the angular momenta defined byEqs. (14)-(16). occ. ε p + j | j z | l | l z | (a) proton 2 -17.4 0.00 1.5 1.5 1.1 1.0neutron 2 -6.6 0.22 1.1 0.6 1.2 0.9(b) proton 2 -14.1 0.08 1.6 1.5 1.2 1.0neutron 2 -5.3 0.98 2.2 0.5 1.8 0.3(c) proton 2 -12.5 0.97 2.2 0.5 2.0 0.2neutron 2 -7.0 0.09 1.8 1.5 1.4 1.0(d) proton 2 -15.6 0.99 2.5 0.5 2.3 0.1neutron 2 -4.4 0.01 2.8 0.5 2.6 0.1(e) proton 2 -16.0 0.00 1.5 1.4 1.1 1.0neutron 1 -3.8 0.99 2.2 0.5 1.8 0.4(f) proton 1 -12.6 0.53 1.9 0.9 1.6 0.8neutron 2 -6.6 0.98 2.1 0.6 1.8 0.3(g) proton 1 -12.4 0.72 2.3 0.9 2.1 0.6neutron 2 -7.2 0.11 1.9 1.4 1.6 1.0(h) proton 1 -13.1 0.52 1.9 1.0 1.6 0.8neutron 2 -8.2 0.92 2.2 0.7 1.9 0.4 negative-parity component as confirmed from the proper-ties of the single-particle orbit listed in Table I. The tablealso shows that two valence neutrons occupy positive-parity orbit ( sd -shell) indicating the 2 ~ ω excitation. Itis noted that a similar configuration, i.e. a triangular 3 α cluster core with 2 ~ ω excited valence neutrons, was alsofound in C [27]. Be( π -bond) Be( σ -bond) { {{ FIG. 3. The schematic figure showing the π and σ -orbitsaround the linear chain. The combination of the p orbitsperpendicular to the symmetry axis generates π orbits, whilethe combination of parallel orbits generates σ orbit. In the strongly deformed region, the linear-chain con-figurations appear. There is an energy plateau aroundthe local energy minimum at ( β, γ ) = (1 . , 0) which ap-proximately corresponds to the deformation ratio equalto 3 : 1. As clearly seen in its density distribution shownin Fig. 2 (c), this local energy minimum has pronounced3 α cluster structure with linear alignment. The prop-erties of the proton single-particle orbit shows that thelast two protons are promoted into sd -shell, that is be-cause of the Pauli principle in the linear-chain configu-ration. The density distribution and properties of thevalence neutron orbits show that they correspond to the π -orbit of the molecular-orbit picture which is schemati-cally illustrated in Fig. 3. Namely, the valence neutronorbit is a linear combination of the p -orbits perpendicu-lar to the symmetry axis and has the angular momenta | j z | = 1 . | l z | = 1 . 0. We call this configuration π -bond linear chain in the following. This property of thevalence neutron orbit is common to that found in C[27]. However, it should be noted that the the π -orbitof C do not have parity symmetric distribution butis localized between the center and right α clusters. Inother words, this configuration is the parity asymmetricand has He+ Be-like structure, which is consistent withthe discussion made in Ref. [24, 33]. Because this linear-chain configuration and the triangular configuration ex-plained above have asymmetric internal structures, weexpect that the corresponding negative-parity partnersmay exist and constitute the inversion doublets.With further increase of the deformation, the otherlinear-chain configuration which we call σ -bond linearchain appears around ( β, γ ) = (1 . , 0) which was notmentioned in Ref. [24]. From the density distribution(Fig. 2 (d)), it is clear that this configuration has an-other valence neutron orbit that correspond to the σ -orbit which is a linear combination of p -orbit parallelto the symmetry axis and has the angular momenta | j z | ≈ . 50 and | l z | ≈ 0. It is interesting to note that thisconfiguration has parity symmetric shape, and hence, donot have its negative-parity partner.The energy minimum of the energy surface for the 1 − states (Fig. 1 (b)) is located at ( β, γ ) = (0 . , ◦ ) withthe binding energy of − . sd -shell from p -shell (1 p h configuration).Because the triangular configuration and the π -bondlinear-chain of the positive parity are parity asymmetric,their counterparts appear in the negative-parity states.Fig. 2 (f) and (g) show the triangular configuration andthe π -bond linear-chain configuration in the negative-parity state located at ( β, γ ) = (0 . , ◦ ) and (1 . , ◦ ),respectively. Although the cluster cores are more dis-torted than the positive-parity states, their neutronsingle-particle configurations are similar to their positive-parity counterparts. However, as explained in the nextsection, these cluster configurations do not form single rotational band and are fragmented into many states be-cause of the mixing with other non-cluster states. For ex-ample, the negative-parity π -bond linear-chain stronglymixes with the strongly deformed non-cluster states suchas the configuration shown in Fig. 2 (h), that makesit impossible to identify the linear-chain rotational stateuniquely. B. Excitation spectrum Figure 4 shows the spectrum of the positive-paritystates obtained by the GCM calculation. The proper-ties of the several selected states are listed in Tab. II.We classified the excited states which have large α re-duced widths as cluster states. The detail of the α re-duced widths is given in the section IV B. In the case ofthe positive-parity states, the cluster states are assignedin the rotational bands without uncertainty, because theband member states are connected by the strong E TABLE II. Excitation energies (MeV) and proton and neu-tron root-mean-square radii (fm) of several selected states.Numbers in the parenthesis are the observed data [40, 41].band J π E x r p r n ground 0 +1 +1 +2 K π = 0 + +2 +1 +3 K π = 2 + +1 +2 π -bond 0 +4 +5 +5 +2 σ -bond 0 +7 +10 +11 The ground state and the first excited state (2 +1 ) aredominantly composed of the configurations around theenergy minimum of the energy surface. The ground statehas the largest overlap with the configuration shown inFig. 2 (a) that amounts to 0.98, and the calculated bind-ing energy is − . − . +1 state is also reasonably described.Owing to its triaxial deformed shape, the triangularconfiguration generates the rotational bands built on the -105-100-95-90-85-80-75-70-65 0 1 2 3 4 5 6 7 8 9 10 He+ Ben+ C He+2 α angular momentum e n e r gy [ M e V ] σ -bond linear chain π -bond linear chainnon-cluster statesexp.ground bandtriangular band FIG. 4. (color online) The positive-parity energy levels up to J π = 10 + . Open boxes show the observed states with the definitespin-parity assignments taken from Ref. [40], and other symbols show the calculated result. The filled circles, triangles andfilled boxes show the ground, triangular and linear-chain bands, while lines show the non-cluster states.TABLE III. The calculated in-band B ( E 2) strengths for thelow-spin positive-parity states in unit of e fm . For thenegative-parity states, the transitions between the low-spinfragmented cluster states (diamonds in Fig. 5) are shown andthe transitions less than 10 e fm are not shown. The numberin parenthesis is the observed data [42]. J i → J f B ( E J i → J f )ground → ground 2 +1 → +1 K π = 0 + +2 → +2 → triangular K π = 0 + +1 → +2 +1 → +2 K π = 2 + +1 → +3 → triangular K π = 2 + +2 → +1 +2 → +3 π -bond linear chain 2 +5 → +4 → π -bond linear chain 4 +5 → +5 +2 → +5 σ -bond linear chain 2 +10 → +7 → σ -bond linear chain 4 +11 → +10 − → − − → − − → − − → − − → − − → − − → − +2 and 2 +3 states that are shown by triangles in Fig. 4. We call them K π = 0 + and 2 + bands, respectively inthe following, although the mixing of the K quantumnumber in their GCM wave functions is not negligible.Compared to the linear-chain states, these bands haveless pronounced clustering and α clusters are consider-ably distorted, therefore the band head energies are wellbelow the cluster thresholds. The member states havelarge overlap with the configuration shown in Fig. 2 (b)which amount to, for example, 0.91 in the case of the 0 +2 state.The linear-chain configurations generate two rotationalbands in Fig. 4. The first one which we call π -bondlinear-chain band is built on the 0 +4 state at 14.6 MeVclose to the α threshold energy and composed of the π -bond linear-chain configurations. The band head state(the 0 +4 state) has large overlap with the configurationshown in Fig.2 (c) which amounts to 0.87. The otherband which we call σ -bond linear-chain band is built onthe 0 +7 state at 22.2 MeV (about 9.18 MeV above the α threshold) and composed of the σ -bond linear-chainconfigurations shown in Fig.2 (d). This intrinsic wavefunction has the largest overlap with the band head statethat amounts to 0.99. The π -bond linear-chain band isthe candidate of the observed resonances and the compar-ison with the observation is discussed in the next section.In the case of the negative-parity states shown in Fig.5, it is found that the α cluster configurations are frag-mented into many excited states. As a result, the E − states that stronglydecay to the 1 − state. Those fragments of the clus-ter configurations are shown by diamond symbols whichare mainly composed of the configurations shown in Fig. -100-95-90-85-80-75-70 1 2 3 4 5 6 7 8 9 10exp.non-cluster states n+ C He+ Be He+2 α cluster states angular momentum e n e r gy [ M e V ] FIG. 5. The negative-parity energy levels up to J π = 10 − . Open boxes show the observed states with the definite spin-parityassignments taken from Ref. [40], and other symbols show the calculated result. The diamonds show the cluster states havingnon-negligible α reduced widths, while lines show the non-cluster states. − state has the largest overlap with theconfiguration shown in Fig. 2 (h) that amounts to 0.93.But, at the same time, this state also have large over-lap with the triangular configuration shown in Fig. 2 (f)and the π -bond linear-chain shown in Fig. 2 (g) whichamounts to 0.85 and 0.70, respectively. This means thatthese fragmented states are the mixture of cluster statesand non-cluster states. The fragmentation of the clusterconfigurations can be more clearly seen in their α reducedwidths which are discussed in the next section. IV. DISCUSSIONA. Excitation energies of the linear-chain bands In this section, we focus on the excitation energies ofthe linear-chain bands and compare them with the ex-periments [20, 29, 30]. The results of the present calcula-tion and the experimental candidates are summarized inFig. 6. By the measurement of the Be( Li,d) C reac-tion, von Oertzen et al. [20] reported a candidate of thelinear-chain band whose band-head energy is below the α threshold energy. Freer et al. [29] and Fritsch et al. [30]independently measured He + Be resonant scatteringusing radioactive Be beam, and reported the candidatesabove the threshold energy. The resonance energies of 4 + state reported by Freer et al and Fritsch et al. are closeto each other, but those of 2 + state differ. However, itmust be kept in mind that the assignment of the 2 + stateby Freer et al. is tentative as mentioned in their report.Then, we see that the calculated energy of the π -bond linear chain is close to the resonances observed in -95-90-85-80 0 2 4 6 exp. [Freer]exp. [Fritsch]exp. [von Oertzen] He+ Be He+2 α angular momentum e n e r gy [ M e V ] π -bond linear chain σ -bond linear chain FIG. 6. (color online) The calculated and observed linear-chain candidates in positive parity. Open boxes show the ob-served data reported by Refs. [20, 29, 30]. Filled boxes showthe energies of the π -bond and σ -bond linear chain states. the He + Be resonant scattering except for the tenta-tively assigned 2 + state. In addition, as discussed in thenext section, the α reduced widths of the π -bond linearchain and those observed resonances are close to eachother. Hence, we conclude that the resonances observedin the He + Be resonant scattering should be the π -bond linear chain. The excited states reported by vonOertzen et al are approximately 5 MeV lower than the π -bond linear chain, and it energetically corresponds tothe triangular band. The measurement of the α widthsof those candidate will make this assignment sure. The σ -bond linear chain is energetically located higher thanany observed resonances and does not have the experi- -94-92-90-88-86-84 1 2 3 4 5 α + Be He+ Be e n e r gy [ M e V ] exp. [Freer]exp. [Fritsch]exp. [von Oertzen]cluster states π -bond linear chainangular momentum FIG. 7. (color online) The calculated cluster states and ob-served linear-chain candidates in negative parity. Open boxesshow the observed data reported by Refs. [20, 29, 30]. Filleddiamonds are the fragmented excited states with cluster con-figuration. Red boxes shows the π -bond linear-chain pro-jected to negative-parity. mental counterpart. As we see later, the σ -bond linearchain is dominantly composed of the He + Be(0 +2 ) and He + Be(2 +2 ) component. Therefore, we consider thatit is not easy to populate this band by ordinary transferreaction or resonant scattering.Fig. 7 summarizes the negative-parity results. In con-trast to the positive parity, there are so many fragmentsof the cluster configurations in the theoretical result. Asa result, the correspondence between the theory and ex-periment is not unique. We also performed an additionaltest calculation. We pickup the π -bond linear-chain con-figuration with positive parity shown in Fig. 2 (c) andartificially project it to the negative-parity to estimatethe energy of the ideal π -bond linear chain with negative-parity. The results is shown by the red filled boxes in Fig.7. We see that the energy of the ideal linear-chain is toohigh to be assigned to the observed resonances. Thus,the present calculation does not support the formationof the linear chain in negative parity. B. Reduced width Figure 8 shows the α reduced widths of several selectedlow-spin states with positive parity. The decay channelsare indicated as [ Be( j π ) ⊗ l ] where j π and l denotethe angular momenta of the Be ground band and therelative motion between Be and α particle, respectively.Here, Be is assumed to have two neutrons in π -orbit.The dominance of the π -bond configuration in the Bewas confirmed by the observations [43].There are two prominent features to be noted in theseresults. The first is the magnitude of the reduced widths. The π -bond linear-chain band (the 0 +4 , 2 +5 , 4 +5 and 6 +2 states) have large reduced widths compared to the tri-angular bands and the ground state. It is also notedthat the α reduced widths of other excited states arealso smaller than the π -bond linear-chain band, and evensmaller than or comparable with the triangular bands.Hence, in the calculated energy region, the π -bond lin-ear chain band has the largest reduced width. In Fig.8(b)-(d), the observed reduced widths of the linear-chaincandidates [29] are also shown for 2 + , 4 + and 6 + states.Since the decay to the Be ground state was assumed in the R-matrix analysis made in Ref. [29], those val-ues may be compared with the calculated results forthe [ Be(0 +1 ) ⊗ l ] channel, and we see that only the π -bond linear-chain band can explain the magnitude ofthe observed reduced widths. Thus, both of the ob-served excitation energies and reduced widths are reason-ably explained by the π -bond linear-chain band, and weconsider that the linear-chain formation in the positive-parity looks plausible.It is also interesting to note that the other linear-chainband, i.e. the σ -bond linear-chain band, has suppressedreduced widths despite of their prominent α clustering.The reason is simple. Because the σ -bond linear-chainband does not have valence neutron in π -orbit, it is or-thogonal to the decay channels to the Be ground statethat has π -orbit neutron. This is confirmed in Fig. 8(e) where the reduced widths for the decays to the Bewith σ -bond (the excited rotational band of Be) areshown. Since other bands do not have valence neutronsin σ -orbit, their reduced widths are suppressed, and onlythe σ -bond linear-chain band has large widths.Another point to be noted is the decay pattern of the π -bond linear-chain band. The reduced widths in the[ Be(2 +1 ) ⊗ l ] channels are as large as or even larger thanthose in the [ Be(0 +1 ) ⊗ l ] channel. This dominance ofthe Be(2 +1 ) component in the π -bond linear-chain bandis owe to its unique structure. When three α particles arelinearly aligned, because of the strong angular correlationbetween α particles, the Be(2 +1 ) and Be(4 +1 ) compo-nents become large. This property is in contrast to theHoyle state where α particles are mutually orbiting with l = 0, and hence, the Be(0 +1 ) component dominates [8].Similar property of the linear-chain configuration werealso discussed in C[34]. Therefore, if the large contam-ination of the Be(2 +1 ) component is confirmed, it willbe a strong evidence for the linear-chain formation.The Be(2 +1 ) component in the triangular bands alsoshow an interesting feature. There are two triangularbands with K π = 0 + and K π = 2 + . The 0 +2 , 2 +2 and 4 +1 states are the member of the K π = 0 + band, while the2 +3 , 4 +2 and 6 +1 are member of the K π = 2 + band. Here,we clearly see that the Be(0 +1 ) component is dominantin the K π = 0 + band, while the Be(2 +1 ) componentis dominant in the K π = 2 + band. This feature is ex-plained by Fig. 9. In the triangular bands, He and Beare placed in a triangular shape and the intrinsic z -axisis chosen to be perpendicular to the deformation axis of Be. Since the K quantum number is the angular mo-mentum directed to the intrinsic z -axis, K must be equalto the angular momentum of Be. This makes the dif-ference in the amount of the Be(2 +1 ) component in the K π = 0 + and 2 + bands.For the negative parity, we show the states which havethe reduced widths larger than 0.1 MeV / in Fig. 8(f)-(h). We can see that there are many states whichhave non-negligible α reduced widths, and not able toidentify the linear-chain band. As already mentioned,the linear-chain configurations are coupled with the non-cluster configurations and fragmented into many statesas found in Ref. [24]. We also see that none of the calcu-lated state can explain the observed reduced widths thatare twice larger than the present results. This requiresfurther theoretical study of the negative-parity states,although the current result looks negative to the linear-chain formation in the negative-parity. V. SUMMARY In order to investigate the existence of the linear-chainstate, we have studied the excited states of C based onthe AMD calculations.In the positive-parity states, we found that three differ-ent configurations appear depending on the magnitude ofthe deformation and the valence neutron configurations.At oblate deformed region, the triangular configurationof 3 α cluster was obtained, while at strong deformed pro-late region, two different linear-chain configurations withvalence neutrons in π -orbit and σ -orbit were obtained.These cluster configurations generate clear rotationalbands. The π -bond linear chain generates a rotationalband around the α threshold energy, while triangularand σ -bond linear chain generate rotational bands well below and well above the threshold. The energy of the π -bond linear chain is in reasonable agreement with theresonances observed by the He + Be, while the trian-gular band is close to the excited states observed by the Be( Li,d) C reaction. The analysis of the α reducedwidth confirms the assignment of the π -bond linear chainto the observed resonances, because the calculated andmeasured widths showed reasonable agreement. Thus,the positive-parity linear-chain formation in C looksplausible. Furthermore, the calculation predicts that the π -bond linear-chain will also decay to the Be(2 +1 ) aswell as to the Be(0 +1 ). This characteristic decay patternwill be, if measured, another evidence of the linear-chainformation.In the negative-parity states, the negative-parity part-ners of the cluster states were also obtained by the en-ergy variation. However, because of the mixing with thenon-cluster configurations, these cluster configurationsare fragmented into many excited states. As a result,many excited states that have sizable α reduced widthare obtained, and it makes the correspondence betweenthe theory and experiment ambiguous. Thus, the presentresult is negative for the linear-chain formation in thenegative-parity, although further studies are in need toidentify the structure of the observed negative-parity res-onances. ACKNOWLEDGMENTS The authors acknowledges the fruitful discussions withDr. Suhara, Dr. Kanada-En’yo, Dr. Fritsch, Mr.Koyama and Dr. H. Otsu. One of the authors (M.K.)acknowledges the support by the Grants-in-Aid for Sci-entific Research on Innovative Areas from MEXT (GrantNo. 2404:24105008) and JSPS KAKENHI Grant No.16K05339. [1] E. Uegaki, S. Okabe, Y. Abe and H. Tanaka, Progr.Theor. Phys. , 1262 (1977); ibid. , 1621 (1979).[2] M.Kamimura, Nucl. Phys. A , 456 (1981).[3] P. Descouvemont and D. Baye, Phys. Rev. C , 54(1987).[4] Y. Kanada-En’yo Phys. Rev. Lett. , 051306 (2003).[7] M. Chernykh, H. Feldmeier, T. Neff, P. von Neumann-Cosel and A. Richter, Phys. Rev. Lett. , 032501 (2007).[8] Y. Funaki, H. Horiuchi, and A. Tohsaki, Prog. Part. Nucl.Phys. , 254 (1956).[10] M. Seya, M. Kohno and S. Nagata, Prog. Theor. Phys. , 204 (1981).[11] W. von Oertzen, Z. Phys. A , 37 (1996); ibid. ,355 (1997). [12] N. Itagaki and S. Okabe, Phys. Rev. C , 044306 (2000).[13] Y. Kanada-En’yo, H. Horiuchi and A. Dot´e, Phys. Rev.C , 064304 (1999).[14] Y. Kanada-En’yo, M. Kimura and H. Horiuchi, C. R.Physique , (2003) 497.[15] Y. Kanada-En’yo, M. Kimura and A. Ono, PTEP ,(2012) 01A202.[16] N. Itagaki, S. Okabe, K. Ikeda and I. Tanihata, Phys.Rev. C (2001) 014301.[17] B. J. Greenhalgh, et al. , Phys. Rev. C (2002) 027302.[18] W. von Oertzen and H. G. Bohlen, C. R. Physique , 465(2003).[19] H. G. Bohlen, et al. , Phys. Rev. C (2003) 054606.[20] W. von Oertzen, et al. , Eur. Phys. J. A (2004) 193.[21] N. I. Ashwood, et al. , Phsy. Rev. C , 064607 (2004).[22] N. Itagaki, W. von Oertzen and S. Okabe, Phys. Rev. C (2006) 067304.[23] D. L. Price et al. , Phys. Rev. C (2007) 014305. [24] T. Suhara and Y. Kanada-En’yo, Phys. Rev. C (2010)044301.[25] J. Maruhn, N. Loebl, N. Itagaki, M. Kimura, Nucl. Phys.A , 1 (2010).[26] N. Furutachi and M. Kimura, Phys. Rev. C (2011)021303(R).[27] T. Baba, Y. Chiba and M. Kimura, Phys. Rev. C ,064319 (2014)[28] P. W. Zhao, N. Itagaki, and J. Meng, Phys. Lett. ,022501 (2015).[29] M. Freer, et al. , Phys. Rev. C , 054324 (2014).[30] A. Fritsch et al. , Phys. Rev. C , 014321 (2016).[31] A. Volkov, Nucl. Phys. , 33 (1965).[32] J. F. Berger, M. Girod, and D. Gogny, Comput. Phys.Comm. (1991) 365.[33] T. Suhara and Y. Kanada-En’yo, Phys. Rev. C ,024328 (2011).[34] Y. Suzuki, H. Horiuchi, and K. Ikeda, Prog. Theor. Phys. , 5 (1972). [35] M. Kimura, Phys. Rev. C (2004) 044319.[36] Y. Kanada-En’yo and H. Horiuchi, Prog. Theor. Phys. , 115 (1995).[37] M. Kimura, R. Yoshida and M. Isaka, Prog. Theor. Phys. , 287 (2012).[38] D. L. Hill and J. A. Wheeler, Phys. Rev. , 1102 (1953).[39] Y. Kanada-En’yo, T. Suhara, and Y. Taniguchi, Prog.Theor. Exp. Phys. , 073D02.[40] F. Ajzenberg-Selove, Nucl. Phys. A , 1 (1991).[41] I. Angelia and K. P. Marinovab, At. Data Nucl. DataTables , 69 (2013).[42] S. Raman, C. W. Nestor Jr., and P. Tikkanen, At. DataNucl. Data Tables , 1 (2001).[43] D. L. Powell, G. M. Crawley, B. V. N. Rao and B. A.Robson, Nucl. Phys. A , 65 (1970). γ α [ M e V / ] γ α [ M e V / ] γ α [ M e V / ] γ α [ M e V / ] [ Be(0 + ) l =0][ Be(2 + ) l =2][ Be(4 + ) l =4] +1 +2 +4 +7 [ Be(0 + ) l =2][ Be(2 + ) l =2] [ Be(4 + ) l =2] +1 +2 +3 +5 +10 [ Be(2 + ) l =0][ Be(2 + ) l =4] [ Be(4 + ) l =4][ Be(4 + ) l =6] exp. [ Be(0 + ) l =6][ Be(2 + ) l =6][ Be(2 + ) l =4][ Be(2 + ) l =8][ Be(4 + ) l =2][ Be(4 + ) l =4][ Be(4 + ) l =6] +1 +2 +6 exp. (d) J=6(e) J=0 (f) J=1(g) J=3 (h) J=5(b) J=2(a) J=0 groundband triangularK=0+ triangularK=0+ triangularK=2+triangularK=2+ σ -bondlinear chain σ -bondlinear chain σ -bondlinear chain π -bondlinear chaingroundband triangularK=0+ σ -bondlinear chain π -bondlinear chain π -bondlinear chain π -bondlinear chain +1 +2 +5 +11 exp. [ Be(0 + ) l =4][ Be(2 + ) l =4] [ Be(4 + ) l =0][ Be(2 + ) l =2][ Be(2 + ) l =6] [ Be(4 + ) l =2][ Be(4 + ) l =4] γ α [ M e V / ] (c) J=4 triangularK=0+ triangularK=2+ σ -bondlinear chain π -bondlinear chain 111 1111 111111 1111 1111111 [ Be(0 + ) l =0][ Be(2 + ) l =2][ Be(4 + ) l =4] +1 +2 +4 +7 (7.5) (14.6) (22.2) (0.0) -3 -5 -7 [ Be(0 + ) l =1][ Be(2 + ) l =3][ Be(4 + ) l =3][ Be(2 + ) l =1][ Be(4 + ) l =5] (17.5) (21.9) (13.6) γ α [ M e V / ] exp. [ Be(0 + ) l =3][ Be(2 + ) l =3][ Be(4 + ) l =3][ Be(2 + ) l =1][ Be(4 + ) l =5][ Be(2 + ) l =5][ Be(4 + ) l =1] -4 -5 -6 -8 -9 -10 γ α [ M e V / ] exp. 5 -4 -2 -3 -7 [ Be(0 + ) l =5][ Be(2 + ) l =3][ Be(4 + ) l =3][ Be(2 + ) l =7][ Be(4 + ) l =5][ Be(2 + ) l =5][ Be(4 + ) l =1] γ α [ M e V / ] FIG. 8. (color online) The calculated α -decay reduced widths compared with the observed widths reported in Ref. [29]. Panels(a)-(d) show the decay of the positive-parity states to the ground band of Be ( π -bonded Be). Panel (e) shows the decay ofthe 0 + states to the excited band of Be ( σ -bonded Be). Panels (f)-(h) show the decay of the negative-parity states to theground band of Be ( π -bonded Be). Numbers in parenthesis show the excitation energy. Be( π -bond) K z FIG. 9. The schematic figure which explain the relationshipbetween the K quantum number and the angular momentumof10