3$\alpha$-cluster structure and monopole transition in $^{12}$C and $^{14}$C

3$\alpha$-cluster structures and monopole transitions of $0^+$ states in $^{12}$C and $^{14}$C were investigated with $3\alpha$- and $^{10}$Be+$\alpha$-cluster models. A gas-like $3\alpha$ state and a bending-chain $3\alpha$ state were obtained in the $0^+_2$ and $0^+_3$ states of $^{12}$C, respectively. In $^{14}$C, a linear-chain 3$\alpha$ structure is found in the $0^+_4$ state near the $^{10}$Be+$\alpha$ threshold, but a cluster gas-like state does not appear because valence neutrons attract $\alpha$ clusters and suppress spatial development of 3$\alpha$ clustering. It was found that the linear-chain state in $^{14}$C is stabilized against the bending and $\alpha$ escaping modes by valence neutrons. The monopole transition strengths in $^{12}$C are enhanced by $3\alpha$-cluster developing, whereas, those in $^{14}$C are not enhanced so much because of the tight binding of $\alpha$ clusters by valence neutrons.


INTRODUCTION
Cluster structure is one of the important aspects in nuclear systems, in particular in light nuclei including unstable nuclei.One of the typical examples of cluster structures is a 3α cluster structure in 12 C.The 3α cluster in 12 C has been intensively investigated theoretically and experimentally [1][2][3][4][5][6][7][8][9][10][11][12][13][14].In the 1950s, a linear-chain 3α state was suggested by Morinaga to describe the 0 + 2 state in 12 C near the 3α threshold energy [22,23].However, this idea was excluded at least for the 0 + 2 state by the experimental α-decay width larger than the expectation for the linear chain state.Later, the 0 + 2 state was understood as a gas-like 3α state [1,5,13].The large monopole transition strength between the ground and 0 + 2 states supports the gas-like 3α structure in the 0 + 2 .Generally, monopole transitions are receiving a lot of attention because they are good probes to identify cluster structures in excited states [15][16][17][18][19][20][21].For 12 C, various cluster structures other than the gas-like state have been also suggested in an energy region higher than the 0 + 2 state.For instance, the bending-chain 3α state was predicted by Antisymmetrized Molecular Dynamics (AMD) and Fermionic molecular dynamics (FMD) calculations [24][25][26].
For neutron-rich C isotopes, possibilities of linear-chain 3α structures have been discussed in many theoretical and experimental works [27][28][29][30][31][32][33][34][35][36].One of the major interests is whether excess neutrons stabilize linear-chain 3α structures in neutron-rich C. For 14 C, the β-γ constraint AMD calculation has predicted a linear-chain 3α structure and its rotational band at an energy slightly higher than the 10 Be+α threshold energy [33].The predicted linearchain 3α structure shows a 10 Be+α cluster structure, in which valence neutrons attract 2α clusters form the 10 Be core and the third α locates at the head-on position of the deformed 10 Be core.In recent experiments, the 2 + and 4 + states have been reported by 10 Be scattering on α and regarded as candidates for members of the predicted linear-chain band in 14 C [36].Also for 16 C, a linear chain structure was predicted by an AMD calculation [34].
Our aim in this paper is to theoretically investigate cluster structures of 0 + states in 12 C and 14 C, and discuss their contributions to monopole transitions.For this aim, we adopt 3αand 10 Be+α-cluster models.We apply the generator coordinate method (GCM) with these models, and investigate 3α dynamics without and with valence neutrons in 12 C and 14 C.In this work, we particularly focus on α-cluster motion around Be cores and α-α motion in the Be cores to discuss appearance and stabilization of gas-like and linear-chain 3α structures.Roles of valence neutrons in 14 C are also discussed.
This paper is organized as follows.In Sec. 2, we explain the present framework of the 3α-and the 10 Be+α-cluster models with the GCM.The calculated results of 12 C and 14 C are shown in Sec. 4. In Sec. 5, we analyze details of cluster structures and discuss their contributions to monopole transition strengths.Finally, in Sec.6, a summary is given.
written as follows, Φ α (R k ; 1, 2, 3, 4) = ϕ p↑ (R k ; 1)ϕ p↓ (R k ; 2)ϕ n↑ (R k ; 3)ϕ n↓ (R k ; 4), where A is the antisymmetrizing operator for all nucleons.Φ α (R k ) is the α-cluster wave function expressed by the harmonic oscillator (0s) 4 configuration with a width parameter ν, which is localized around the position R k .χ σ and τ σ are the spin and isospin parts of the single-particle wave function.In this work, we focus on the motion of the third α cluster (α 3 ) around the 2α, therefore, we rewrite the generator coordinates R 1 , R 2 , R 3 with the α-α distance (d 2α ) and the distance (D α ) and angle (θ α ) of the α 3 position (D α ) relative to the center of mass position of the 2α as
To calculate the energy spectra and monopole transitions in 12 C, we superpose the parity and total angular-momentum projected wave functions as where P J± M K is the parity and total-angular-momentum projection operator.The coefficients c 0 + n are determined by diagonalizing Hamiltonian and norm matrices.

10 Be + α-cluster model
In order to describe cluster structures in 14 C, we perform a GCM calculation of 3α-nn by extending the 3α-cluster GCM.In this paper, we pay a particular attention to 10 Be + αcluster structures in 14 C, and therefore, the nn configuration is optimized to describe the 10 Be cluster in 14 C as follows.We start from the Brink-Bloch 3α-cluster wave function Φ 3α (R 1 , R 2 , R 3 ) described previously.We express two valence neutrons with Gaussian wave packets as ϕ ↑n (R n↑ )ϕ ↓n (R n↓ ).Here, the Gaussian centers R n↑ and R n↓ for spin-up and down neutrons are chosen so that the subsystem of 2α + nn describes a 10 Be cluster as, where two neutrons have the Gaussian centers with the same real part (position) and the opposite imaginary part (momentum).This wave function for the 10 Be cluster is in principle equivalent to the simplified cluster model proposed by Itagaki et al. [38], in which the parameter λ for the finite momentum in opposite directions for spin-up and down neutrons is introduced to gain the spin-orbit interaction.d indicates the distance of nn from the 2α and λ is the parameter for the momentum of Gaussian wave packets.For each α-α distance , we optimize the parameters d and λ, so as to minimize the subsystem energy 10 Be(0 + ) and use the optimized values in the calculation of 14 C. φ specifies the angle of the nn position around the 2α, namely, that from the 3α plane in the 14 C wave function (Fig. 1 (b)).By using the 10 Be wave function Φ10 Be (R 1 , R 2 , φ), we express the basis wave function for 14 C as where the recoil effect is taken into account as 4R 1 + 4R 2 + 4R 3 + R n↑ + R n↓ = 0.In a similar way to the 3α-cluster wave function, we rewrite the generator coordinates R 1 , R 2 , R 3 with d 2α for the α-α distance and D α and θ α for the α 3 position and express the total wave function as Φ10 Be+α (d 2α , D α , θ α , φ).
To calculate the energy spectra and monopole transitions in 14 C, we superpose the parity and total angular-momentum projected wave functions by treating four parameters d 2α , D α , θ α , φ as the generator coordinates as

Isoscalar monopole transition strengths
The monopole operator M(IS0) is defined as where r i is the coordinate of the ith particle and R c.m. is the total center of mass coordinate.The monopole transition strength from the ground state to the excite state 0 + n is calculated by the squared matrix element of M(IS0) as

MODEL PARAMETER AND EFFECTIVE INTERACTIONS
In the present calculation, we use the same width parameter ν = 0.235 fm −2 used in Ref. [33].
For For effective interactions, we adopt the same interactions used in Ref. [33] for the study of 14 C with the β-γ constraint AMD (βγ-AMD).We use the Volkov No.2 [39] with W = 1 − M , B = H = 0.125, and M = 0.60 for the central force, the spin-orbit term in the G3RS [40] with u 1 = −u 2 = 1600 MeV, and the Coulomb force approximated by seven Gaussians.Here we discuss energies of subsystems, the 8 Be and 10 Be clusters, calculated using the 2α-and 2α+nn-cluster models, respectively.Figure 2 shows the d 2α dependence of the 0 + energy for 8 Be projected from the 2α wave function, and that for 10 Be projected from Φ10 In the 0 + energy of 8 Be, the energy minimum exists at d 2α = 3.8 fm and the energy curve is found to be very soft toward a large d 2α region.On the other hand, the 0 + energy of 10 Be shows the energy minimum at d 2α = 2.6 fm and rapidly increases in a large d 2α region.It indicates that valence neutrons make 2α to be bound tightly and suppress the α-α distance in 10 Be compared with 8 Be.We calculate energy spectra of 10 Be using the GCM within the present 2α+nn cluster model by superposing the wave functions Φ10 Be (R 1 , R 2 , φ) with the parameter d 2α = 2, 3, 4 fm and check that the present model reproduces well the spectra of 10 Be(0 + 1 ),  Let us discuss the 0 + -projected energy of the 3α wave function Φ 3α (d 2α , D α , θ α ). Figure 3 shows the 0 + -projected energy of the 3α wave function plotted on the X-Z plane for the α 3 position around the 2α with α-α distances d 2α = 2 fm and 4 fm.Here, X and Z are defined as (X, Z) = (D α sin θ α , D α cos θ α ), and the first and second α clusters are located on the Z axis (Fig. 1 (a)).In the energy surface for d 2α = 2 fm, we find a deep energy pocket around (X, Z) = (3, 0) fm which corresponds to the ground state configuration of 12 C. Along the Z axis, the energy is rather high in the D α < 4 fm region because of the Pauli blocking effect from another α cluster.In the D α ∼ 5 fm region, an energy valley is seen from θ α = 90 • toward θ α = 0 • corresponding to the soft bending mode.In the energy surface for d 2α = 4 fm, the energy surface shows only a shallow minimum around (X, Z) = (2.5, 0) fm and an energy plateau is spread widely toward the large D α region.It can be seen that the energy does not increase so much with the increase of d 2α and D α indicating that the energy is very soft against the spatial development of the 3α cluster except for the energy pocket at the compact 3α configuration for the ground state. 12C.  12 C obtained by the Full-GCM calculation of the 3α-cluster model measured from the 3α threshold energy.Experimental spectra [41] and theoretical ones of RGM [3], GCM [1], βγ-AMD [9] and OCM [8] are also shown.

GCM calculation of
We superpose the 3α wave functions and obtain the ground and excited 0 + states of 12 C with the 3α GCM calculation described in Sec. 2. Figure 4 shows the obtained 0 + energy spectra compared with the experimental data.Theoretical spectra calculated using the 3α resonating group method (RGM) by Kamimura et al. [3], the 3α GCM by Uegaki et al. [1] the β-γ constraint antisymmetrized molecular dynamics (βγ-AMD) by Suhara et al. [9], and the 3α orthogonality condition model (OCM) by Kurokawa et al. [8] are also shown.These calculations are microscopic 3α-cluster model calculations except for the OCM and are in principle consistent with the present model but there are slight differences in the interaction parameters and finite volume sizes.In the present calculation, we obtain the ground state with the compact 3α cluster structure, the 0 + 2 state with the gas-like 3α structure, and the 0 + 3 state with the banding-chain 3α structure.These three states are consistent with those of the 3α GCM by Uegaki et al.
Experimentally, 12 C(0 + 1 , 0 MeV) and 12 C(0 + 2 , 7.65 MeV) are well known.Around the excitation energy E x ∼ 10 MeV, two 0 + states at 9.04 MeV and 10.8 MeV have recently been reported [42].However, in the 3α GCM and βγ-AMD calculations, only one 0 + state is obtained in the E x ∼ 10 MeV region, which can be assigned to the experimental 12 C(0 + , 10.8 MeV).The 3α OCM calculation by Kurokawa et al. [8] and the resent 3α calculation by Funaki et al. [13] predicted another 0 + state of a higher nodal state of the 0 + 2 state, which may be assigned to the experimental 12 C(0 + , 9.04 MeV).In the present calculation, we obtain the 0 + 3 state which is likely to be assigned to the experimental 12 C(0 + , 10.8 MeV) state.Above the 0 + 3 state, we obtain the 0 + 4 state with the higher nodal feature, which might be assigned to 12 C(0 + , 9.04 MeV).However, the present model space of D α ≤ 7 fm is not enough to obtain a converged result for the 0 + 4 state and could overestimate its excitation energy.3 ) obtained by the Full-GCM calculation of the 3α-cluster model.Overlaps are plotted on the X-Z plane for the α 3 position.
To discuss 3α cluster structures of the obtained 0 + states in detail, we calculate the overlap between the GCM wave function for 12 C(0 + n ) and the basis 3α wave function defined as where | P 0 + Φ 3α is normalized to be P 0 + Φ 3α | P 0 + Φ 3α = 1. Figure 5 shows the overlap plotted on the X-Z plane for the α 3 position around the 2α (d 2α = 3 fm), which indicates the α 3 probability distribution in 12 C(0 + n ).In the ground state, the α 3 probability is concentrated around (X, Z) = (2, 1) fm with the maximum value 91 % showing the compact 3α-cluster structure.In 12 C(0 + 2 ), the α 3 probability is spread widely in the large D α region in both cases of d 2α = 2 and 4 fm, indicating the cluster gas-like nature of freely moving 3α clusters like a gas.This is consistent with the results reported in prior researches [14].For 12 C(0 + 3 ), the α 3 probability distribution shows a large amplitude in 5 fm < Z < 7 fm region near the Z-axis corresponding to the aligned 3α configuration, but the probability is spread in the finite X region with the maximum amplitude at (X, Z) = (2.5, 6) fm.Note that the second peak exists around (X, Z) = (5, 0) fm with the amount of 20%.These results indicate the bending-chain 3α configuration rather than the linear-chain structure.
In Table 1, monopole transition strengths from the ground state and root-mean-square (rms) radii for 12 C(0 + n ) are shown.The gas-like state ( 12 C(0 + 2 )) has the remarkably large monopole transition strength and rms radius.The 0 + 3 state of 12 C also has the significant monopole transition strength though it is less than half of that of 12 C(0 + 2 ).The present calculation overestimates the experimental B(IS0) by about factor 2 mainly because of the adopted effective interaction.In the present calculation, we use the original Volkov No.2 interaction as explained previously.The 3α GCM calculation using the Volkov No.1 gives a reasonable B(IS0) value [1], and the 3α RGM using the Volkov No.2 with a modified strength  We discuss the energy of the 10 Be+α wave function Φ10 Be+α (d 2α , D α , θ α , φ) with respect to the α 3 position around the 10 Be core.The 0 + -projected energy is calculated for the K = 0 projected 10 Be core ( 10 Be(K = 0)+α wave function) as where the angle φ for the nn position at eight points φ = π 8 , 3π 8 , 5π 8 , • • • , 15π 8 are summed with the equal weight to project the 10 Be core to the K = 0 eigen state.Figure 6 shows the 0 + energy plotted on the X-Z plane for the α 3 position around the 10 Be(K = 0) core with d 2α = 2 fm and 4 fm.In the energy surface for d 2α = 2 fm, we find the energy minimum around (X, Z) = (2, 0) fm which corresponds to the ground state configuration of 14 C. Along the Z axis, the energy is relatively high in the D α < 3.5 fm region because of the Pauli blocking effect from another α cluster similarly to 12 C.In the D α ∼ 4 fm region, an energy valley is seen toward θ α = 0 • , which corresponds to the linearly aligned 3α on the Z axis.Comparing the energy surfaces for 14 C with those for 12 C, we find significant differences.The energy pocket for the ground state is deeper and the energy valley exists in the smaller D α region in 14 C than in 12 C.It means that valence neutrons effectively give an additional attraction between α clusters and keep three clusters in a compact region compared with the 3α system without valence neutrons.7 0 + energy spectra of 14 C obtained by the Full-GCM calculation of the 10 Be+αcluster model measured from the 10 Be+α threshold energy.The experimental data [43] and the theoretical spectra of the βγ-AMD [33] are also shown.

GCM calculation of 14 C
We perform the 10 Be + α GCM calculation and obtain the ground and excited 0 + states of 14 C by superposing the 10 Be+α wave functions.Figure 7 shows the calculated 0 + energy spectra of 14 C compared with the experimental data as well as the theoretical results of Ref. [33] calculated using the βγ-AMD by Suhara et al.In the present calculation, we obtained the 0 + 4 state having the dominant linear-chain structure slightly above the 10 Be+α threshold energy.It corresponds to the linear-chain state predicted as the 0 + 5 state with the βγ-AMD.Below this state, we obtain only two excited 0 + states.Compared with the βγ-AMD calculation, an excited 0 + state below the linear-chain state is missing in the present calculation, maybe, because α cluster breaking is omitted in the present model space of the 10 Be+α wave functions.
Experimentally, only three states below the 10 Be+α threshold energy have been confirmed to be 0 + states.The ground state is overbound from the 10 Be+α threshold in the present calculation similarly to the βγ-AMD calculation.This overbinding problem might come from the effective interaction of the two-body density-independent central force used in the two calculations.Fig. 8 Overlaps with the 10 Be+α wave functions (d 2α = 3 fm) for 14 C(0 + 1 ), 14 C(0 + 2 ), 14 C(0 + 3 ), and 14 C(0 + 4 ) obtained by the Full-GCM calculation of the 10 Be+α model.
To measure the α 3 probability around the 10 Be, we calculate the overlap between the obtained GCM wave function for 14 C(0 + n ) and the K = 0 projected 10 Be+α cluster wave function as Figure 8 shows the α 3 probability in 14 C(0 + n ) plotted on the X-Z plane for the α 3 position around the 10 Be(K = 0) with d 2α = 3 fm.The 14 C ground state shows the compact structure with the large α 3 probability around (X, Z) = (1.5, 0.75) fm.In 14 C(0 + 2 ), the α 3 probability is distributed in the larger D α region than 14 C(0 + 1 ), but the spatial development of clustering is not as remarkable as 12 C(0 + 2 ). 14C(0 + 3 ) has only small overlap with the 10 Be(K = 0)+α wave function meaning that major component of this state is not the 10 Be(K = 0)+α configuration. 14C(0 + 4 ) has the large α 3 probability near the Z-axis in 3 fm < Z < 5 fm region.The largest probability exists around (X, Z) = (0, 4.5) fm corresponding to the linear-chain 3α structure.In comparison with 12 C, the probability in 14 C(0 + 4 ) is more concentrated on the Z-axis, i.e., the fluctuation against the bending motion is weaker than 12 C(0 + 3 ).In the present GCM calculation, we take into account the α-cluster motion around the 10 Be core, which is not sufficiently considered in the work by Suhara et al. [33].Nevertheless, we obtain the linear-chain 3α structure having the cluster structure quite similar to that predicted in Ref. [33].The linear-chain structure in 14 C is stable against the bending mode as well as the escaping mode of the α cluster owing to the existence of valence neutrons.On the other, we find no excited state showing the gas-like feature of 3α in 14 C.The reason is that α clusters are more strongly attracted by valence neutrons in 14 C than 12 C. Below the linear-chain state, we obtain 14 C(0 + 2 ) and 14 C(0 + 3 ) with characters different from 12 C(0 + 2 ).More detailed discussions are given in the next section.
In Table 2, we show the monopole transition strengths and rms radii of 14 C(0 + n ).We do not find remarkable monopole transitions for excited 0 + states in 14 C as that of 12 C(0 + 2 ).Among 14 C(0 + 2,3,4 ), 14 C(0 + 4 ) has significant monopole transition strength comparable to that of 12 C(0 + 3 ).The calculated rms radius of the ground state reasonably reproduces the experimental radius.The rms radii of the excited 0 + states 14 C(0 + 2,3,4 ) are longer than 14 C(0 + 1 ) but they are smaller than those of 12 C(0 + 2,3 ).This indicates again that valence neutrons attract α clusters and suppress the spatially development of the 3α clustering.

DISCUSSION
In this section, we discuss details of cluster modes in excited 0 + states of 12 C and 14 C such as the α 3 -cluster rotation (θ α ) mode around Be cores and the α-α (d 2α ) mode in Be cores and their effects on the monopole transition strengths by performing truncated GCM calculations within restricted model spaces.
5.1.Effects of α 3 -cluster rotation around Be cores: analysis of fixed-θ GCM To discuss effects of the α 3 -cluster rotation around the 8 Be core in 12 C, we preform the GCM calculation without the α 3 -cluster rotation by fixing θ α = π/2 of the 3α wave functions (named fixed-θ GCM).Namely, we use only two parameters, d 2α and D α , as the generator coordinates in the fixed-θ GCM.Note that the angular motion of the α 3 cluster is equivalent to the 8 Be core rotation in a J π eigen state.We compare the fixed-θ GCM (without rotation) with the Full-GCM (with rotation).In Fig. 9, we show the 0 + energy spectra and monopole transition strengths of 12 C calculated by the fixed-θ GCM compared with the Full-GCM.We also show squared overlaps of the wave functions for the 0 + states between two calculations.
It is found that the 0 + 1 state of the fixed-θ GCM (fixed-θ 0 + 1 ) has 100% overlap with that of the Full-GCM (Full-GCM 0 + 1 ) indicating that the α 3 rotation gives almost no contribution to the ground state.On the other hand, the rotation effect gives important contributions to excited 0 + states.The fixed-θ 0 + 2 has 73% overlap with the Full-GCM 0 + 2 and gains significant energy by the rotation to form the gas-like state.For the Full-GCM 0 + 3 , there is no corresponding state in the fixed-θ GCM because the bending chain 3α structure is mainly described by θ ∼ 0 configurations, which are omitted in the fixed-θ GCM.It should be noted that the fixed-θ 0 + 2 has 23% overlap with the Full-GCM 0 + 3 .In other words, the component of the fixed-θ 0 + 2 is split by the rotation into the Full-GCM 0 + 2 and 0 + 3 states with fractions of 73% and 23%, respectively.
Let us compare the monopole transition strengths of 12 C between fixed-θ GCM and Full-GCM calculations.In the fixed-θ GCM, the monopole transition strength is concentrated at the 0 + 2 state.The strong monopole transition for the fixed-θ 0 + 2 is split by the rotation into the Full-GCM 0 + 2 and 0 + 3 , consistently with the split of the fixed-θ 0 + 2 component to these two states with the fractions of 73% and 23%.As a result, in the Full-GCM result, 12 C(0 + 2 ) has the remarkable monopole transition strength, whereas 12 C(0 + 3 ) has the relatively smaller but significant monopole transition strength.The present result is consistent with the rotation effect in the monopole strength discussed by one of the authors in Ref. [14].In a similar way to the analysis for 12 C, we calculate the fixed-θ GCM for 14 C to discuss the rotation effect of the α 3 cluster around the 10 Be core.In the fixed-θ GCM for 14 C, we use three parameters, d 2α , D α , and φ, as the generator coordinates and fix θ α = π/2.In Fig. 10, we show the 0 + energy spectra and monopole transition strengths of 14 C calculated by the fixed-θ GCM compared with the Full-GCM, and also show the squared overlaps of the obtained wave functions between two calculations.For 14 C(0 + 1 ), the α 3 rotation gives almost no contribution (100% overlap).For 14 C(0 + 2 ), the fixed-θ 0 + 2 has 90% overlap with 13 the Full-GCM 0 + 2 and gains about 1 MeV energy by the rotation.For 14 C(0 + 3 ) of the Full-GCM, the fixed-θ 0 + 3 has the dominant overlap as 60%.For the linear-chain state, 14 C(0 + 4 ) obtained by the Full-GCM, there is no corresponding state in the fixed-θ GCM because it contains dominantly the θ ∼ 0 components, which are omitted in the fixed-θ GCM.It should be noted that the fixed-θ 0 + 3 component is fragmented by the rotation into the Full-GCM 0 + 3 and 0 + 4 states with fractions of 60% and 30%, respectively.In the monopole transition strengths calculated by the fixed-θ GCM, the strength is concentrated at the 0 + 3 state at E x ∼ 20 MeV, but the absolute value of B(IS0) for this state is about half of the fixed-θ 0 + 2 state of 12 C.As a result of rotation, the monopole strength concentrated at the fixed-θ 0 + 3 is fragmented into the Full-GCM 0 + 3 and 0 + 4 in this energy region.However the ratio of the monopole strengths for the Full-GCM 0 + 3 and 0 + 4 is not consistent with the fractions (60% and 30%) of the fixed-θ 0 + 3 component in these two states.Namely, even though the Full-GCM 0 + 3 contains the significant fixed-θ 0 + 3 component, it has the relatively weaker monopole transition than the Full-GCM 0 + 4 .In order to understand cluster structures and their relation to the monopole strengths in 14 C, further detailed analysis is necessary by taking into account valence neutron configurations.

Valence neutron configurations in 14 C
In this section, we discuss the effect of valence neutron mode (φ) in the cluster structures of 14 C.As described previously, we adopt 8 points φ = π/8, 3π/8, 5π/8, 7π/8, 9π/8, 11π/8, 13π/8, 15π/8 for the nn angle position around the 2α in the 10 Be core.To see valence neutrons distribution in the 10 Be core, we classify the neutron configurations into two groups.One is the set φ = π/8, 7π/8, 9π/8, 15π/8 called vertical, in which two neutrons are located in a direction vertical to the X-Z plane, and the other is the set φ j = 3π/8, 5π/8, 11π/8, 13π/8 called planar, in which the 3α and two neutrons form approximately planar configurations on the X-Z plane.Here, the 10 Be(Vertical) + α and 10 Be(Planar) + α wave functions of vertical and planar configurations projected onto 0 + states are defined as We calculate the 10 Be(vertical) + α and 10 Be(planar) + α components in the 0 + states of 14 C obtained by the Full-GCM wave function.In Fig. 11, the vertical and planar components in 14 C(0 + n ) are shown on the X-Z plane for the α 3 position around the 10 Be core with d 2α = 3 fm.We find the vertical component is dominant in the ground state because the planar configuration is suppressed in such the compact cluster state because of the Pauli blocking from the α 3 cluster for valence neutrons in the planar configuration.On the other hand, the planar configuration is dominant in 14 C(0 + 2 ).This state has a somewhat enhanced cluster structure, in which the valence neutrons in the planar configurations gain much potential energy compared with those in the vertical configurations.This result is consistent with the discussion in Ref. [33].In 14 C(0 + 3 ), the vertical component is dominant whereas the planar one is minor.We find the linear-chain 3α state at 14 C(0 + 4 ) has almost the same amount of the vertical and planar components in the region near the Z-axis.However, 14 C(0 + 4 ) also has 14 ) Planar Fig. 12 Monopole transition strengths from the approximated ground state wave function of 14 C to the 10 Be(K = 0) + α wave functions, those to the 10 Be(Vertical) + α wave functions, and those to the 10 Be(Planar) + α wave functions.
the significant (about 20%) vertical component around (X, Z) = (4, 0) fm.As shown later, this vertical component contributes to the monopole strength of 14 C(0 + 4 ).Let us discuss which configurations of Φ 0 + 10 Be(K=0)+α , Φ 0 + 10 Be(vertical)+α and Φ 0 + 10 Be(planar)+α are directly excited by the monopole operator from the ground state.We calculate the monopole transition from 14 C(0 + 1 ) to a single In Fig. 12, the monopole strengths in 14 C for the three kinds of configurations are plotted on the X-Z plane.We find that the vertical configurations in the θ α ∼ π/2 region are directly excited by the monopole operator from the ground state, because the ground state dominantly consists of compact vertical configurations with θ α ∼ π/2 as shown in Fig. 11.
Since the monopole operator does not cause rotation, it cannot excite planar configurations.It means that 14 C(0 + n ) are excited by the monopole operator mainly through the vertical component in the θ α ∼ π/2 region.
Let us consider an additional truncation of the model space in the GCM calculation to discuss effects of valence neutron configurations in the 10 Be core on the cluster structures and monopole strengths in 14 C. We perform the GCM calculation using only the vertical basis wave functions (V-GCM), in which we superpose the vertical wave functions Φ 0 Fig. 13 Overlaps with the 10 Be(vertical)+α wave function for 0 + 1 , 0 + 2 , 0 + 3 , and 0 + 4 obtained by the V-GCM calculation.The overlaps are plotted on the X-Z plane for the α 3 position around the 10 Be(vertical) core with d 2α = 3 fm.
In order to discuss 3α dynamics in the model space of the vertical configuration, we show, in Fig. 13, the overlaps with the 10 Be(Vertical)+α wave function on the X − Z plane for the 0 + states obtained by the V-GCM calculation.The 0 + 1 state has a compact structure.In the 0 + 2 state, the α 3 probability is spread widely in the larger D α region than the 0 + 1 state and has a gas-like feature, although the spatial development of clustering is not as prominent as 12 C(0 + 2 ) because of the attraction from the valence neutrons in the vertical 16 configuration.The 0 + 3 state shows a bending-chain 3α structure similar to 12 C(0 + 3 ), but the cluster development is weaker than that of 12 C(0 + 3 ).Fig. 14 0 + energy spectra and monopole transition strengths of 14 C obtained by the fixed-θ V-GCM and V-GCM calculations.The energy levels with significant ( 30%) overlap between two calculations are connected by dotted lines.
To discuss effects of the α 3 -cluster rotation around the 10 Be(vertical) core, we preform the fixed-θ GCM calculation in the vertical configurations, called fixed-θ V-GCM and compare the result with the V-GCM result.Figure 14 shows the 0 + energy spectra and monopole transition strengths of 14 C calculated by the fixed-θ V-GCM and the V-GCM as well as the overlap of the obtained wave functions.
The 0 + 1 state of the fixed-θ V-GCM has 100% overlap with that of the V-GCM.The 0 + 2 state obtains about 2 MeV energy gain by the α 3 rotation.For the 0 + 3 state of the V-GCM, there is no corresponding state in the fixed-θ V-GCM, because it has a bending-chain 3α structure like that of 12 C(0 + 3 ) and contains dominantly θ α ∼ 0 configurations, which are omitted in the fixed-θ V-GCM.It should be commented that the component of the fixed-θ V-GCM 0 + 2 is fragmented by the rotation into the V-GCM 0 + 2 and 0 + 3 states with fractions of 64% and 25% .
In the monopole strengths, the strongly concentrated strength can be seen at the 0 + 2 state in the fixed-θ V-GCM, and it is split by the rotation to the 0 + 2 and 0 + 3 states in the V-GCM.The ratio of the monopole strengths of the V-GCM 0 + 2 and 0 + 3 states is consistent with the ratio of the component of the fixed-θ V-GCM 0 + 2 state in these two states.Namely, as a result of the dominant fixed-θ V-GCM 0 + 2 component in the V-GCM 0 + 2 , the V-GCM 0 + 2 has the largest monopole strength.These results indicate that features of the cluster structures and monopole strengths in 14 C in the vertical model space are quite similar to those of 12 C, though some quantitative differences are seen in the weaker cluster development and monopole transitions in 14 C of the V-GCM than 12 C.It means that 3α clusters with the valence neutrons in the vertical configuration form cluster structures similar to the 3α system without valence neutrons except for the somewhat weaker spatial extend of cluster motion because of the additional attraction by the valence neutrons in the vertical configuration.Therefore, the qualitative differences of cluster features of the Full-GCM between 14 C and 12 C originate in coupling of the vertical with planar configurations.We compare the V-GCM result with the Full-GCM one to discuss the coupling effect of the vertical and planar configurations.Figure 15 shows the 0 + energy spectra and monopole transition strengths of 14 C of the V-GCM and Full-GCM as well as the overlap of the obtained wave functions.The coupling gives almost no contribution to the ground state.The V-GCM 0 + 2 state is split into two states (0 + 2 and 0 + 3 ) in the Full-GCM because of the coupling with the planar configurations.The linear-chain state at 14 C(0 + 4 ) of the Full-GCM has the major overlap of 84% with the V-GCM 0 + 3 state having the bending-chain 3α structure.It means that the bending-chain 3α structure is stabilized to form the linear chain 3α structure because of the coupling of the vertical and planar configurations.
In the monopole transition strengths, the V-GCM 0 + 2 has the strongest transition reflecting the large 10 Be(Vertical) + α component in the θ α ∼ π/2 region, which can be directly excited by the monopole operator from the ground state as discussed previously.However, the coupling with the planar configurations fragments, the monopole transition strengths of the V-GCM 0 + 2 state into the Full-GCM 0 + 2 and 0 + 3 states, and also somewhat enhances the monopole transition strength of the Full-GCM 0 + 4 state.As a result, there is no concentration of the monopole strengths in this energy region. 14C(0 + 4 ) of the Full-GCM has the monopole strength comparable to 12 C(0 + 3 ) because it contains the significant 10 Be(Vertical) + α component in the θ α ∼ π/2 region, which is regarded as the remains of the bending-chain structure in the fixed-θ V-GCM 0 + 3 .

Effects of α-α mode of 8 Be and 10 Be cores
As discussed in Sec.4.1, the 8 Be core is soft, but the 10 Be core is relatively stiff against the α-α mode.To see the effect of the α-α mode of the Be cores in 12 C and 14 C, we perform the GCM calculations with a fixed α-α distance, d 2α = 3 fm (named fixed-d 2α GCM).In Fig. 16, we show the 0 + energy spectra and monopole strengths in 12 C and 14 C calculated by the fixed-d 2α GCM compared with those of the Full-GCM.We find significant energy gains in 12 C(0 + 2 ), 12 C(0 + 3 ), and 12 C(0 + 4 ) because of the α-α mode in the 8 Be core.Moreover, the monopole strength for 12 C(0 + 2 ) is remarkably enhanced by the soft α-α mode.In contrast to the significant effects of the α-α mode in 12 C, the effects of the α-α mode in 14 C is not so remarkable, the fixed-d 2α GCM results are qualitatively similar to the Full-GCM for 14 C.It is concluded that the α-α mode plays an important role in the lowered energy and the enhanced monopole strength of 12 C(0 + 2 ), whereas it is less important in 14 C.

Cluster features of 12 C and 14 C
As shown previously, in 12 C, the 0 + 2 shows the cluster gas-like feature, and the 0 + 3 has the bending-chain structure.In 14 C, the linear-chain structure is obtained in the 0 + 4 state, but a cluster gas-like structure does not appear in low-lying 0 + states.In the previous sections, of the 3α clustering compared with the 3α-cluster structure in 12 C.The valence neutrons stabilize the linear-chain state in 14 C against the bending mode and the escaping mode of the α cluster.
We also investigated monopole transitions in 12 C and 14 C and analyzed effects of α-cluster motion and nn configurations on the monopole transitions.In 14 C, because of the strong coupling of the α-cluster motion and nn configurations, monopole transition strengths are fragmented into many 0 + states.The monopole transition strengths in 12 C are enhanced by the α-α motion in the 8 Be core, but however, those in 14 C are not enhanced so much by the α-α motion in the 10 Be core because valence neutrons tightly bind 2α clusters in the core.

Fig. 1
Fig. 1 Schematic figures for the 3α-and 10 Be + α-cluster models.d 2α indicates the α-α distance, and D α and θ α indicate the α 3 position relative to the center of mass position of the 2α.In the right panel for the 10 Be + α-cluster model, φ indicates the angle of the nn position on the X-Y plane.

Fig. 2 0
Fig. 2 0 + -projected energies of 8 Be and 10 Be obtained using the 2α and 2α + nn wave functions, respectively.The energies are shown as functions of the α-α distance d 2α .

Fig. 9 0
Fig.90 + energy spectra and monopole transition strengths of12 C obtained by the fixedθ and Full-GCM calculations.The energy levels with significant ( 30%) overlaps between two calculations are connected by dotted lines.

Fig. 10 0
Fig.100 + energy spectra and monopole transition strengths of 14 C obtained by the fixedθ and Full-GCM calculations.The energy levels with significant ( 30%) overlap between two calculations are connected by dotted lines.

Fig. 15 0
Fig.150 + energy spectra of 14 C and monopole transition strengths obtained by the V-GCM and Full-GCM calculations.The energy levels with significant ( 30%) overlap between two calculations are connected by dotted lines.

Fig. 16 0
Fig.160 + energy spectra and monopole strengths of12 C and 14 C calculated by the fixed-d 2α GCM compared with those of the Full-GCM.The energy levels with significant ( 30%) overlap between two calculations are connected by dotted lines.
It means that we use only three values of d 2α for the 10 Be + α GCM calculation to save numerical costs because d 2α dependence of the 2α + nn subsystem energy shows a deep minimum around d 2α = 3 fm due to valence neutrons.More details are described later.

Table 2
[43]pole transition strengths and rms radii for 0 + states of 14 C. Experimental data are from Ref.[43].