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The coexistence of various structures, such as diverse shapes and cluster structures, is a fundamental property of atomic nuclei. In neutron-rich nuclei, a core$+n$ structure can compete with nuclear deformation due to the small neutron separation energy. A neutron-rich carbon isotope, $^{17}$C, exemplifies the appearance of the deformation and the core+$n$ decoupling in its spectrum, which is desirable for a deeper understanding of the coexistence phenomena in neutron-rich nuclei. We aim to describe and understand this coexistence phenomenon in the low-lying levels of $^{17}$C in a unified manner considering explicitly the degrees of freedom of both the quadrupole deformation and the relative motion between a $^{16}$C core and a valence neutron. Method: We adopt the generator coordinate method (GCM) with the antisymmetrized molecular dynamics (AMD) to describe various configurations. We superpose various basis wave functions generated by the energy variation by imposing two types of constraints: the $\beta$-$\gamma$ and the $d$ constraints, which control the degree of the quadrupole deformation and the relative motion between a $^{16}$C core and a valence neutron, respectively. We find that the experimental energy level is well reproduced by the present method, including both deformed and $^{16}$C+$n$ configurations. The ground $3/2^{+}$ and second excited $5/2^{+}$ states exhibit a triaxially deformed shape, while the main component of the first excited $1/2^{+}$ state is a $^{16}$C($0^{+}$) core plus an $s$-wave neutron configuration. The tail of the valence neutron is significantly improved by including the $^{16}$C+$n$ basis functions explicitly. The explicit inclusion of both the quadrupole deformation and the relative motion between a core and a valence neutron is essential to describe the coexistence phenomena observed in neutron-rich nuclei in the AMD+GCM framework., Comment: 8 pages, 5 figures

Background: Nuclear deformation provides a crucial characteristic of nuclear structure. Conventionally, the quadrupole deformation length of a nucleus, $\delta_{2}$, has often been determined based on a macroscopic model through a deformed nuclear potential with the deformation length $\delta^{\rm (pot)}_{2}$, which is determined to reproduce the nuclear scattering data. This approach assumes $\delta_{2}=\delta^{\rm (pot)}_{2}$ although there is no theoretical foundation. Purpose: We clarify the relationship between $\delta_{2}$ and $\delta^{\rm (pot)}_{2}$ for high-energy heavy-ion scattering systematically to evaluate the validity of the conventional approach to determine the nuclear deformation. Method: The deformation lengths for the $^{12}$C inelastic scattering by $^{12}$C, $^{16}$O, $^{40}$Ca, and $^{208}$Pb targets at $E/A$ = 50--400 MeV are examined. First, we perform microscopic coupled-channel (CC) calculations to relate $\delta_{2}$ of the deformed density into the inelastic scattering cross section. Second, we use the deformed potential model to determine $\delta^{\rm (pot)}_{2}$ so as to reproduce the microscopic CC result. We then compare $\delta^{\rm (pot)}_{2}$ with $\delta_{2}$. Results: We find that $\delta^{\rm (pot)}_{2}$ is about 20--40 \% smaller than presumed $\delta_{2}$, showing strong energy and target dependence. Further analysis, which considers higher-order deformation effects beyond the derivative model, reveals that $\delta^{\rm (pot)}_{2}$ is still about 15--35 \% smaller than $\delta_{2}$. Conclusion: Our results suggest that one needs to be careful when the deformed potential model for the high-energy heavy-ion scattering is used to extract the nuclear deformation. The conventional approach may underestimate the deformation length $\delta_2$ systematically., Comment: 7 pages, 5 figures

Recently, we have determined surface distributions of $\alpha$ clusters in the ground state of $^{20}\mathrm{Ne}$ from $\alpha$-transfer cross sections, without investigating the properties of its excited states. In this paper we extend our comprehension of $\alpha$-cluster structures in excited states of nuclei through reaction studies. In particular we focus on $^{16}\mathrm{O}$, for which attention has been paid to advances of structure theory and assignment regarding $4^+$-resonance states. We study the surface manifestation of the $\alpha$-cluster states in both the ground and excited states of $^{16}\mathrm{O}$ from the analysis of the $\alpha$-transfer reaction $^{12}\mathrm{C}(^6\mathrm{Li},d)^{16}\mathrm{O}$. The $\alpha$-transfer reaction is described by the distorted-wave Born approximation. We test two microscopic wave functions as an input of reaction calculations. Then a phenomenological potential model is introduced to clarify the correspondence between cluster-wave functions and transfer-cross sections. Surface peaks of the $\alpha$-wave function of $^{16}\mathrm{O}(0^+)$ are sensitively probed by transfer-cross sections at forward angles, while it remains unclear how we trace the surface behavior of $^{16}\mathrm{O}(4^+)$ from the cross sections. We are able to specify that the $\alpha$-cluster structure in the $0_1^+$ and $0_2^+$ states prominently manifests itself at the radii $\sim 4$ and $\sim 4.5$~fm, respectively. It is remarkable that the $4_1^+$ state has the $^{12}\mathrm{C}+\alpha$-cluster component with the surface peak at the radius $\sim 4$ or outer, whereas the $^{12}\mathrm{C}+\alpha$-cluster component in the $4_2^+$ state is found not to be dominant. The $4_2^+$ state is difficult to be interpreted by a simple potential model assuming the $^{12}\mathrm{C}+\alpha$ configuration only., Comment: 14 pages, 7 figures

We formulate the "tensor-optimized high-momentum antisymmetrized molecular dynamics (TO-HMAMD)" framework for ab initio calculations of nuclei by hybridizing the tensor-optimized (TO-) and high-momentum (HM-) AMD approaches. This hybrid approach has advantages in both analytical simplicity and numerical efficiency comparing with other AMD-based methods which treat the bare interaction, especially for heavier nuclear systems. In this work, the $s$-shell nucleus $^{4}$He is calculated with TO-HMAMD by including up to double product of nucleon-nucleon ($NN$) correlations, described by using high-momentum pairs and spatial correlation functions of nucleons. The total energy and radius of $^{4}$He nucleus are well reproduced using the AV8$^\prime$ interaction. The spin-isospin channel dependence is also discussed for $NN$-correlations, which are found to be mostly contributed in the even-state channels, especially the triplet-even channel. Analyses of the analytical formation and numerical results suggest that TO-HMAMD could be a promising framework for general nuclear systems., Comment: 8 figures

Many-body correlations play an essential role in the ab initio description of nuclei with nuclear bare interactions. We propose a new framework to describe light nuclei by the hybridization of the tensor-optimized antisymmetrized molecular dynamics (TOAMD) and the high-momentum AMD (HM-AMD), which we call "HM-TOAMD". In this framework, we describe the many-body correlations in terms of not only the correlation functions in TOAMD, but also the high-momentum pairs in the AMD wave function. With the bare nucleon-nucleon interaction AV8', we sufficiently reproduce the energy and radius of the 3H nucleus in HM-TOAMD. The effects of tensor force and short-range repulsion in the bare interaction are nicely described in this new framework. We also discuss the convergence in calculation and flexibility of the model space for this new method., Comment: 9 pages, 5 figures, 1 table, Accepted by PTEP

We treat the tensor correlation in antisymmetrized molecular dynamics (AMD) including large-relative-momentum components among nucleon pairs for finite nuclei. The tensor correlation is described by using large imaginary centroid vectors of Gaussian wave packets for nucleon pairs with opposite directions, which makes a large relative momentum. We superpose the AMD basis states, in which one nucleon pair has various relative momenta for all directions; this new method is called "high-momentum AMD" (HM-AMD). We show the results for $^4$He using the effective interaction having a strong tensor force. It is found that HM-AMD provides a large tensor matrix element comparable to the case of the tensor-optimized shell model (TOSM), in which the two-particle-two-hole (2p-2h) excitations are fully included to describe the tensor correlation. The results of two methods agree with each other at the level of the Hamiltonian components of $^4$He. This indicates that in HM-AMD the high-momentum components described by the imaginary centroid vectors of the nucleon pair provide the equivalent effect of the 2p-2h excitations for the tensor correlation., Comment: 11 pages, 4 figures, added references

We developed a new variational method for tensor-optimized antisymmetrized molecular dynamics (TOAMD) for nuclei. In TOAMD, the correlation functions for the tensor force and the short-range repulsion are introduced and used in the power series form of the wave function, which is different from the Jastrow method. Here, nucleon pairs are correlated in multi-steps with different forms, while they are correlated only once including all pairs in the Jastrow correlation method. Each correlation function in every term is independently optimized in the variation of total energy in TOAMD. For $s$-shell nuclei using the nucleon-nucleon interaction, the energies in TOAMD are better than those in the variational Monte Carlo method with the Jastrow correlation function. This means that the power series expansion using the correlation functions in TOAMD describes the nuclei better than the Jastrow correlation method., Comment: 7 pages, 4 figures. arXiv admin note: text overlap with arXiv:1706.09560, update references

We recently proposed a new variational theory of "tensor-optimized antisymmetrized molecular dynamics" (TOAMD), which treats the strong interaction explicitly for finite nuclei [T. Myo et al., Prog. Theor. Exp. Phys. 2015, 073D02 (2015)]. In TOAMD, the correlation functions for the tensor force and the short-range repulsion and their multiple products are successively operated to the AMD state. The correlated Hamiltonian is expanded into many-body operators by using the cluster expansion and all the resulting operators are taken into account in the calculation without any truncation. We show detailed results for TOAMD with the nucleon-nucleon interaction AV8$^\prime$ for $s$-shell nuclei. The binding energy and the Hamiltonian components are successively converged to exact values of the few-body calculations. We also apply TOAMD to the Malfliet-Tjon central potential having a strong short-range repulsion. TOAMD can treat the short-range correlation and provided accurate energies of $s$-shell nuclei, reproducing the results of few-body calculations. It turns out that the numerical accuracy of TOAMD with double products of the correlation functions is beyond the variational Monte Carlo method with Jastrow's product-type correlation functions., Comment: 25 pages, 24 figures, corrected typos

Tensor-optimized antisymmetrized molecular dynamics (TOAMD) is the basis of the successive variational method for nuclear many-body problem. We apply TOAMD to finite nuclei to be described by the central interaction with strong short-range repulsion, and compare the results with the unitary correlation operator method (UCOM). In TOAMD, the pair-type correlation functions and their multiple products are operated to the AMD wave function. We show the results of TOAMD using the Malfliet-Tjon central potential containing the strong short-range repulsion. Adding the double products of the correlation functions in TOAMD, the binding energies are converged quickly to the exact values of the few-body calculations for s-shell nuclei. This indicates the high efficiency of TOAMD for treating the short-range repulsion in nuclei. We also employ the s-wave configurations of nuclei with the central part of UCOM, which reduces the short-range relative amplitudes of nucleon pair in nuclei to avoid the short-range repulsion. In UCOM, we further perform the superposition of the s-wave configurations with various size parameters, which provides a satisfactory solution of energies close to the exact and TOAMD values., Comment: 8 pages, 9 figures

We study the tensor-optimized antisymmetrized molecular dynamics (TOAMD) as a successive variational method in many-body systems with strong interaction for nuclei. In TOAMD, the correlation functions for the tensor force and the short-range repulsion and their multiples are operated to the AMD state as the variational wave function. The total wave function is expressed as the sum of all the components and the variational space can be increased successively with the multiple correlation functions to achieve convergence. All the necessary matrix elements of many-body operators, consisting of the multiple correlation functions and the Hamiltonian, are expressed analytically using the Gaussian integral formula. In this paper we show the results of TOAMD with up to the double products of the correlation functions for the s-shell nuclei, 3H and 4He, using the nucleon-nucleon interaction AV8'. It is found that the energies and Hamiltonian components of two nuclei converge rapidly with respect to the multiple of correlation functions. This result indicates the efficiency of TOAMD for the power series expansion in terms of the tensor and short-range correlation functions., Comment: 7 pages, 5 figures, added references, corrected typos

We develop a new formalism to treat nuclear many-body systems using bare nucleon-nucleon interaction. It has become evident that the tensor interaction plays important role in nuclear many-body systems due to the role of the pion in strongly interacting system. We take the antisymmetrized molecular dynamics (AMD) as a basic framework and add a tensor correlation operator acting on the AMD wave function using the concept of the tensor-optimized shell model (TOSM). We demonstrate a systematical and straightforward formulation utilizing the Gaussian integration and differentiation method and the antisymmetrization technique to calculate all the matrix elements of the many-body Hamiltonian. We can include the three-body interaction naturally and calculate the matrix elements systematically in the progressive order of the tensor correlation operator. We call the new formalism "tensor-optimized antisymmetrized molecular dynamics"., Comment: 36 pages

Direct evidence of the $\alpha$-cluster manifestation in bound states has not been obtained yet, although a number of experimental studies were carried out to extract the information of the clustering. In particular in conventional analyses of $\alpha$-transfer reactions, there exist a few significant problems on reaction models, which are insufficient to qualitatively discuss the cluster structure. We aim to verify the development of the $\alpha$-cluster structure from observables. As the first application, we plan to extract the spatial information of the cluster structure of the $^{20}$Ne nucleus in its ground state through the cross section of the $\alpha$-transfer reaction $^{16}$O($^6$Li,~$d$)$^{20}$Ne. For the analysis of the transfer reaction, we work with the coupled-channel Born approximation (CCBA) approach, in which the breakup effect of $^6$Li is explicitly taken into account by means of the continuum-discretized coupled-channel method based on the three-body $\alpha + d + {}^{16}$O model. The two methods are adopted to calculate the overlap function between $^{20}$Ne and $\alpha + {}^{16}$O; one is the microscopic cluster model (MCM) with the generator coordinate method, and the other is the phenomenological two-body potential model (PM). We show that the CCBA calculation with the MCM wave function gives a significant improvement of the theoretical result on the angular distribution of the transfer cross section, which is consistent with the experimental data. Employing the PM, it is discussed which region of the cluster wave function is probed on the transfer cross section. It is found that the surface region of the cluster wave function is sensitive to the cross section. The present work is situated as the first step in obtaining important information to systematically investigate the cluster structure., Comment: 9 pages, 7 figures

To clarify the effects of $\alpha$-cluster breaking on 3$\alpha$ cluster structures in $^{12}$C, we investigate $^{12}$C using a hybrid model that combines the Brink-Bloch cluster model with the $p_{3/2}$ subshell closure wave function. We have found that $\alpha$-cluster breaking caused by spin-orbit force significantly changes cluster structures of excited $0^{+}$ states through orthogonality to lower states. Spatially developed cluster components of the $0^{+}_{2}$ state are reduced. The $0^{+}_{3}$ state changes from a vibration mode in the bending motion of three $\alpha$ clusters to a chain-like 3$\alpha$ structure having an open triangle configuration. As a result of these structure changes of $0^{+}$ states, the band assignment for the $2^{+}_{2}$ state is changed by the $\alpha$-cluster breaking effect. Namely, in model calculations without the $\alpha$-cluster breaking effect, the $0^{+}_{2}$ state is assigned to be the band-head of the $2^{+}_{2}$ state. However, when we incorporate $\alpha$-cluster breaking caused by the spin-orbit force, the $0^{+}_{3}$ state is regarded as the band-head of the $2^{+}_{2}$ state., Comment: 11 pages, 7 figures

We reanalyze $2\alpha+t$ cluster features of $3/2^-$ states in $^{11}$B by investigating the $t$ cluster distribution around a $2\alpha$ core in $^{11}$B, calculated with the method of antisymmetrized molecular dynamics (AMD). In the $3/2^-_3$ state, a $t$ cluster is distributed in a wide region around $2\alpha$, indicating that the $t$ cluster moves rather freely in angular as well as radial motion. From the weak angular correlation and radial extent of the $t$ cluster distribution, we propose an interpretation of a $2\alpha+t$ cluster gas for the $3/2^-_3$ state. In this study, we compare the $2\alpha+t$ cluster feature in $^{11}$B($3/2^-_3$) with the $3\alpha$ cluster feature in $^{12}$C($0^+_2$), and discuss their similarities and differences., Comment: 14 pages, 11 figures

We propose a simple method to approximately evaluate reduced width amplitude (RWA) of a two-body spinless cluster channel using the norm overlap with the Brink-Bloch cluster wave function at the channel radius. The applicability of the present approximation is tested for the $^{16}$O+$\alpha$ channel in $^{20}$Ne as well as the $\alpha$+$\alpha$ channel in $^8$Be. The approximation is found to be reasonable to evaluate the RWA for states near the threshold energy and it is useful to estimate the $\alpha$-decay width of resonance states. The approximation is also applied to $^9$Li, and the partial decay width of the $^6$He($0^+_1$)+$t$ channel is discussed., Comment: 20 pages, 10 figures

Gamov-Teller transitions from the $^{14}$N ground state to the $^{14}$C ground and excited states were investigated, based on the model of antisymmetrized molecular dynamics. The calculated strengths for the allowed transitions to the $0^+$, $1^+$, and $2^+$ states of $^{14}$C were compared with the experimental data measured by high-resolution charge-exchange reactions. The calculated GT transition to the $2^+_1$ state is strong while those to the $0^+_{2,3}$ and $2^+_{2,3}$ states having dominant $2\hbar\omega$ excited configurations are relatively weak. The present calculation can not describe the anonymously long life time of $^{14}$C, though the strength of the $^{14}$C ground state is somewhat suppressed because of the cluster (many-body) correlation in the ground states of $^{14}$C and $^{14}$N., Comment: 10 pages, 7 figures

We propose an improved version of Antisymmetrized Quasi-Cluster Model (AQCM) to describe a smooth transition from the $\alpha$-cluster wave function to the $jj$-coupling shell model wave function and apply it to the ground state of $^{12}$C. The cluster-shell transition in $^{12}$C is characterized in AQCM by only two parameters: $R$ representing the distance between $\alpha$ clusters and the center of mass, and $\Lambda$ describing the break of $\alpha$ clusters. The optimal AQCM wave function for the ground state of $^{12}$C is an intermediate state between the three-$\alpha$ cluster state and the shell model state with the $p_{3/2}$ subshell closure configuration. The result is consistent with that of the Antisymmetrized Molecular Dynamics (AMD), and the optimal AQCM wave function quantitatively agrees with the AMD one, although the number of degrees of freedom in AQCM is significantly fewer., Comment: 11 pages, 6 figures

Structures of excited states in $^{11}$B are investigated with a method of $\beta$-$\gamma$ constraint antisymmetrized molecular dynamics in combination with the generator coordinate method. Various excited states with developed cluster core structures are suggested in positive- and negative-parity states. For negative-parity states, we suggest a band with a $2\alpha+t$ cluster structure. This band starts from the $3/2^{-}_{3}$ state and can correspond to the experimental band observed recently. In positive-parity states, two $\alpha$ core cluster structures with surrounding nucleons are found. A $K^\pi=1/2^+$ band is suggested to be constructed from a remarkably developed cluster structure with a large prolate deformation. We discuss features of the cluster structure in association with molecular orbital structures of $^{10}$Be., Comment: 13 pages, 9 figures. arXiv admin note: text overlap with arXiv:1004.4954

We propose a method to obtain adiabatic internuclear potentials via energy variation with the intercluster-distance constraint. The adiabatic $^{16}$O + $^{16,18}$O potentials obtained by the proposed method are applied to investigate the effects of valence neutrons in $^{16}$O + $^{18}$O sub-barrier fusions. Sub-barrier fusion cross sections of $^{16}$O + $^{18}$O are enhanced more compared to those of $^{16}$O + $^{16}$O because of distortion of valence neutrons in $^{18}$O., Comment: 11 pages, 5 figures

Cluster states in $^9$Li are investigated with calculations of a $^6$He-$t$ cluster model. Results suggest $^6$He-$t$ cluster states near the $^6$He-$t$ threshold energy. These states construct a $K^\pi=1/2^-$ band and their neutron configuration is similar to that of the $K^\pi=0^+_2$ band in $^{10}$Be., Comment: 7 pages, 9 figures

Two-neutron densities obtained from microscopic wave functions of $^6$He and $^8$He are investigated to reveal di-neutron correlations. In particular, the comparison of the two-neutron density with the product of one-neutron densities is useful for a quantitative discussion of di-neutron correlations. The calculations show that the S=0 spatial two-neutron correlation increases at the surface of $^6$He$(0^+_1)$ and $^8$He$(0^+_2)$. The enhancement is remarkable in the $^6$He$(0^+_1)$ ground state but not as prominent in the $^8$He$(0^+_1)$ ground state. Configuration mixing of many Slater determinants is essential to describe the di-neutron correlations. Two-neutron densities in $^{12}$C wave functions with $\alpha$-cluster structures are also studied., Comment: 12 figures

We investigate the linear-chain structures in highly excited states of $^{14}$C using a generalized molecular orbital model, by which we incorporate an asymmetric configuration of three $\alpha$ clusters in the linear-chain states. By applying this model to the $^{14}$C system, we study the $^{10}$Be+$\alpha$ correlation in the linear-chain state of $^{14}$C. To clarify the origin of the $^{10}$Be+$\alpha$ correlation in the $^{14}$C linear-chain state, we analyze linear $3\alpha$ and $3\alpha+n$ systems in a similar way. We find that a linear 3$\alpha$ system prefers the asymmetric 2$\alpha+\alpha$ configuration, whose origin is the many body correlation incorporated by the parity projection. This configuration causes an asymmetric mean field for two valence neutrons, which induces the concentration of valence neutron wave functions around the correlating 2$\alpha$. A linear-chain structure of $^{16}$C is also discussed., Comment: 13 pages, 8 figures

Structures of excited states in $^{14}$C are investigated with a method of $\beta$-$\gamma$ constraint antisymmetrized molecular dynamics in combination with the generator coordinate method. Various excited states with the developed $3\alpha$-cluster core structures are suggested in positive- and negative-parity states. In the positive-parity states, triaxial deformed and linear-chain structures are found to construct excited bands. Interestingly, $^{10}$Be+$\alpha$ correlation is found in the cluster states above the $^{10}$Be+$\alpha$ threshold energy., Comment: 17 pages, 9 figures

We propose a new method of the $\beta$-$\gamma$ constraint for quadrupole deformation in antisymmetrized molecular dynamics (AMD) to describe various cluster and shell-model structures in the ground and excited states of light nuclei. We apply this method to N=6 isotones, $^{10}$Be, $^{12}$C, $^{9}$Li, and $^{11}$B, and find various structures as functions of the deformation parameters, $\beta$ and $\gamma$. In these nuclei, shell-model-like structures appear in the small $\beta$ region, while cluster structures develop well in the large $\beta$ region where various geometric configurations of clusters are obtained depending on the $\gamma$ parameter. For $^{10}$Be and $^{12}$C, we superpose the basis AMD wave functions obtained by the $\beta$-$\gamma$ constraint method to calculate energy spectra, and prove the advantages of the present method of the two-dimensional $\beta$-$\gamma$ constraint in the framework of AMD., Comment: 23pages, 14 figures

Two-neutron correlation in the $^1S$ channel in quasi two-dimensional (2D) neutron systems at zero temperature is studied by means of the BCS theory with finite-range effective nuclear forces. The dineutron correlation in low density neutron systems confined in an infinite slab is investigated in a simplified model that neutron motion of one direction is frozen. When the slab is thin enough, two neutrons form a tightly bound dineutron with a small size in the quasi-2D system, and a Bose dineutron gas is found in low density limit. With increase of Fermi momentum, the neutron system changes from the Bose-gas phase to the superfluid Cooper pair phase. The density dependence of the neutron pairing shows the BCS-BEC crossover phenomena at finite low-density region. In the transition region, the size shrinking of neutron pair and enhancement of pairing gap are found. The relation to dineutron correlation at surface of neutron-rich nuclei is also discussed., Comment: 18 pages, 10 figures