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The nonlocalized clustering approach is generalized to $$^{8-11}$$ Be isotopes with isospin dependent spin–orbit potential. A new form of the Tohsaki-Horiuchi-Schuck-Ropke (THSR) wave function is introduced to provide a correct description for the $$\sigma $$ -binding neutron in $$^{11}$$ Be. Systematic calculations for $$^{8-11}$$ Be isotopes are performed and results fit well with experimental values. The low energy spectrum of $$^{11}$$ Be is also obtained, especially the correct spin-parity $$1/2^{+}$$ is reproduced for the intruder ground state. The exotic neutron halo structure of $$^{11}$$ Be is studied by calculations of root-mean-square radii and density distribution. The spectroscopic factor is also calculated and discussed for the $$1/2^+$$ ground state of $${}^{11}$$ Be.

We present a microscopic calculation of $\alpha$-cluster formation in heavy nuclei $^{104}$Te ($\alpha$+$^{100}$Sn), $^{212}$Po ($\alpha$+$^{208}$Pb) and their neighbors $^{102}$Sn, $^{102}$Te, $^{210}$Pb and $^{210}$Po by using the quartetting wave function approach. Improving the local density approximation, the shell structure of the core nucleus is considered, and the center-of-mass (c.o.m.) effective potential for the quartet is obtained self-consistently from the shell model wavefunctions. The $\alpha$-cluster formation and decay probabilities are obtained by solving the bound-state of the c.o.m. motion of the quartet and the scattering state of the formed $\alpha$-cluster in the Gurvitz approach. Striking shell effects on the $\alpha$-cluster formation probabilities are analyzed for magic numbers 50, 82 and 126. The computed $\alpha$-decay half-lives of these special nuclei are compared with the newest experimental data., Comment: 16 pages, 4 figures

Quartetting ( $$\alpha $$ -like clustering) occurs in low-density matter ( $${\le }0.03$$ fm $$^{-3}$$ ) which exists, e.g., at the surface of nuclei. It is of interest for the $$\alpha $$ preformation to calculate the $$\alpha $$ decay of heavy nuclei such as $$^{212}$$ Po, but also in light nuclei (e.g., $$^{20}$$ Ne) which shows strong signatures of quartetting. We analyze the intrinsic structure of the $$\alpha $$ -like cluster and the center-of-mass motion of the quartet, in particular the role of Pauli blocking. The Thomas–Fermi model for the (daughter) core nucleus is improved introducing quasiparticle nucleon states. Calculations performed for harmonic oscillator basis states show that the effective potential for the quartet center-of-mass motion remains nearly constant within the core nucleus. The relation to the (Tohsaki–Horiuchi–Schuck–Ropke) THSR approach is discussed.

In order to study the nucleon-nucleon pairing effects in clustering nuclei, we formulate a superposed Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave function, which includes both molecular-orbit and pairing configurations explicitly. With this new wave function, we investigate the anomalous deuteronlike $pn$-pairing effect in $^{10}\mathrm{B}$ with $T=0$ and $S=1$ (isoscalar) by comparing with isovector $NN$ pairs $(T=1,S=0)$ in $^{10}\mathrm{Be}$ and $^{10}\mathrm{C}$. Energies are calculated for the ground states of $^{10}\mathrm{Be},^{10}\mathrm{B}$ and $^{10}\mathrm{C}$ nuclei, and the ${1}_{1}^{+}0$ excited state of $^{10}\mathrm{B}$. These energies are essentially improved comparing with studies using a previous version of THSR wave function. Furthermore, overlaps between the total wave function and the pairing component indicate that the $NN$ pairing effect is more visible in $^{10}\mathrm{B}$ than in $^{10}\mathrm{Be}$ and $^{10}\mathrm{C}$. By analyzing the energies and the overlaps between wave function components, we observe two different mechanisms enhancing the formation of deuteronlike pairs in $^{10}\mathrm{B}$. We also discuss the pairing effect by showing average distances between components in each nucleus and density distributions of valence nucleons.

We focus on the raison d’être of the $\alpha$-chain states on the basis of the fully microscopic framework, where the Pauli principle among all the nucleons is fully taken into account. Our purpose is to find the limiting number of $\alpha$ clusters on which the linear $\alpha$-cluster state can stably exist. How many $\alpha$ clusters can stably make an $\alpha$-chain state? We examine the properties of equally separated $\alpha$ clusters on a straight line and compare its stability with that on a circle. We also confirm its stability in terms of binary and ternary disintegrations including $\alpha$-decay and fission modes. For the effective nucleon–nucleon interaction we employ the F1 force, which has finite-range three-body terms and guarantees overall saturation properties of nuclei. This interaction also gives a reasonable binding energy and size of the $\alpha$ particle, and the $\alpha$–$\alpha$ scattering phase shift. The result astonishes us because we can point out the possible existence of $\alpha$-chain states with vast numbers of $\alpha$ clusters.

The first ${3}^{\ensuremath{-}}$ and ${4}^{\ensuremath{-}}$ states of $^{12}\mathrm{C}$ are studied in the present container model, in which the shift parameter is introduced to break the parity symmetry for projecting out the negative-parity states. Taking the limit as the shift parameter approaches zero and by variational calculations for one-deformed size parameter, the local energy minima are obtained for the ${3}^{\ensuremath{-}}$ and ${4}^{\ensuremath{-}}$ states. It is found that the obtained single Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave functions for ${3}^{\ensuremath{-}}$ and ${4}^{\ensuremath{-}}$ states are 96% and 92% equivalent to the corresponding generator coordinate method (GCM) wave functions, respectively. The calculated intrinsic densities further show that these negative-parity states of three clusters, different from the traditional understanding of a rigid triangle structure, are found to have nonlocalized clustering structure in the two-dimensional container picture.

We study the property of $\ensuremath{\alpha}$ aggregates distributed on Archimedean solids within a microscopic framework, which takes full account of the Pauli principle. We pay special attention to the Coulomb energy for such exotic shapes of nuclei, and we discuss the advantage of $\ensuremath{\alpha}$ clusters with geometric configurations compared with the uniform density distributions in reducing the repulsive effect. We consider four kinds of configurations of $\ensuremath{\alpha}$ clusters distributed on Archimedean solids: dual polyhedra composed of a dodecahedron and an icosahedron, an octacontahedron, and two types of truncated icosahedrons, that is, two kinds of Archimedean solids. The latter two are an icosidodecahedron and a fullerene shape. Putting an $\ensuremath{\alpha}$ cluster on each vertex of the polyhedra, four $\ensuremath{\alpha}$-cluster aggregates correspond to the following four nuclei: Gd (64 protons), Po (84 protons), Nd (60 protons), and a nucleus with 120 protons, respectively.

In order to study the correlations between clusters and nucleons in light nuclei, we formulate a new superposed Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave function which describes both spatially large spreading and cluster-correlated dynamics of valence nucleons. Using this new THSR wave function, the binding energy of $^{9}\mathrm{B}$ is significantly improved in comparison with our previous studies. We calculate the excited states of $^{9}\mathrm{B}$ and obtain an energy spectrum of $^{9}\mathrm{B}$ which is consistent with the experimental results. This includes the prediction of the first $1/{2}^{+}$ excited state of $^{9}\mathrm{B}$ which is not yet fixed experimentally. We study the proton dynamics in $^{9}\mathrm{B}$ and find that the cluster-proton correlation plays an essential role for the proton dynamics in the ground state of $^{9}\mathrm{B}$. Furthermore, we discuss the density distribution of the valence proton with special attention to its tail structure. Finally, the resonance nature of excited states of $^{9}\mathrm{B}$ is illustrated comparing root-mean-square radii between the ground and excited states.

The anomalous large radii are exotic phenomena observed around the neutron dripline. Around the neutron dripline, the weak binding of the last bound neutron(s) causes the drastic increase of the radius, which is called neutron halo structure. Although the nucleus $^{24}$O is located at the dripline of Oxygen isotopes, the separation energies of one and two neutron(s) are 4.19 MeV and 6.92 MeV, respectively. In spite of this enough binding, the enhancement of the matter radius is observed. In this study, we microscopically describe the structure change of $^{22}$O core in $^{24}$O and explain the observed large radius based on the cluster model. Two degrees of freedom for the large radius; the relative distances among four $\alpha$ clusters and size of each $\alpha$ cluster are examined, where Tohsaki interaction, which has finite range three-body terms is employed. The nucleus $^{24}$O has the almost the same amount of the clusterization compared with $^{22}$O, but the expansion of each $\alpha$ cluster plays an important role. When two neutrons are added to $^{22}$O at the center, the expansion of each $\alpha$ clusters is energetically more favored than enhancing the clustering for reducing the kinetic energy of the neutrons. The calculated rms matter radius of $^{22}$O and $^{24}$O are 2.75 fm and 2.92 fm, respectively. Although these are slightly smaller than the experimental values, the jump at $^{24}$O from $^{22}$O is reproduced.

We study $\ensuremath{\alpha}$-cluster structure based on the geometric configurations with a microscopic framework, which takes full account of the Pauli principle, and which also employs an effective internucleon force including finite-range three-body terms suitable for microscopic $\ensuremath{\alpha}$-cluster models. Here, special attention is focused upon the $\ensuremath{\alpha}$ clustering with a hollow structure; all the $\ensuremath{\alpha}$ clusters are put on the surface of a sphere. All the platonic solids (five regular polyhedra) and the fullerene-shaped polyhedron coming from icosahedral structure are considered. Furthermore, two configurations with dual polyhedra, hexahedron-octahedron and dodecahedron-icosahedron, are also scrutinized. When approaching each other from large distances with these symmetries, $\ensuremath{\alpha}$ clusters create certain local energy pockets. As a consequence, we insist on the possible existence of $\ensuremath{\alpha}$ clustering with a geometric shape and hollow structure, which is favored from Coulomb energy point of view. Especially, two configurations, that is, dual polyhedra of dodecahedron-icosahedron and fullerene, have a prominent hollow structure compared with the other six configurations.

The original 3α THSR (Tohsaki-Horiuchi-Schuck-Ropke) wave function was extended to 2α+α THSR wave function for studying the rich cluster structures of 0+ states in 12C, which includes two size parameters and the 2α correlation was included in the container picture. It was found that the extended single THSR wave functions give a very high-accuracy description of the ground state and also the excited 0+ states in 12C. By superposing the 2α+α THSR basis with the radius constraint method, we performed the GCM calculations for the 0+ states. The existence of the 03+ and 04+ states around 10 MeV were confirmed in this microscopical cluster model. The further study indicates the 03+ state can be considered as a breathing-like excited state of the Hoyle state.

The tensor contribution can be directly incorporated in the cluster model in a simplified way. In conventional $\ensuremath{\alpha}$ cluster models, the contribution of the noncentral interactions exactly cancels because of the antisymmetrization effect and spatial symmetry of $\ensuremath{\alpha}$ clusters. The mixing of breaking components of $\ensuremath{\alpha}$ clusters to take into account the spin-orbit and tensor effects is needed. Previously, we proposed a simplified method to include the spin-orbit effect, and also for the tensor part, a simplified model to directly take into account the contribution of the tensor interaction (called SMT) was introduced; however, the contribution of the tensor interaction was quite limited. Here we improve SMT, which is called $i\mathrm{SMT}$. Using newly proposed $i\mathrm{SMT}$, the contribution of the tensor interaction in $^{4}\mathrm{He}$ is more than $\ensuremath{-}40$ MeV, four times larger than the previous version. The method is applied to four-$\ensuremath{\alpha}$ cluster structure of $^{16}\mathrm{O}$. In $^{16}\mathrm{O}$, the tensor contribution is also large, and this is coming from the finite size effect for the distances among $\ensuremath{\alpha}$ clusters with a tetrahedral configuration.

In 1954, Fred Hoyle made a prediction regarding the existence of a resonant state of carbon-12 that would be associated with a triple-alpha process which would be fundamental in nucleosynthesis in stars and would explain the amount of carbon-12 in the Universe. This prediction was spectacularly confirmed by experiments. Nevertheless, the precise properties of the Hoyle state are still the subject of investigation. This Colloquium describes the current theoretical and experimental investigation of this important prediction.

The existence of the ${0}_{3}^{+}$ and ${0}_{4}^{+}$ states around 10 MeV excitation energy in $^{12}\mathrm{C}$ is confirmed by a fully microscopic $3\ensuremath{\alpha}$ cluster model. Firstly, a generator coordinate method (GCM) calculation is performed by superposing optimized $2\ensuremath{\alpha}+\ensuremath{\alpha}$ Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave functions with the radius-constraint method. The obtained two excited ${0}^{+}$ states above the Hoyle state are consistent with the recently observed states by experiment. Secondly, a variational calculation using the single $2\ensuremath{\alpha}+\ensuremath{\alpha}$ THSR wave function orthogonalized to the ground and Hoyle states is made and it also supports the existence of the ${0}_{3}^{+}$ state obtained by the GCM calculation. The analysis of the obtained ${0}_{3}^{+}$ state is made by studying its $2\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ reduced width amplitude, its $2\ensuremath{\alpha}$ correlation function, and the large monopole matrix element between this state and the Hoyle state, which shows that this ${0}_{3}^{+}$ state is a breathing-like excited state of the Hoyle state. This character of the ${0}_{3}^{+}$ state is very different from the ${0}_{4}^{+}$ state which seems to have a bent-arm $3\ensuremath{\alpha}$ structure.

We extend the concept of nonlocalized clustering to the nucleus Be10 with proton number Z=4 and neutron number N=6 (N=Z+2). The Tohsaki-Horiuchi-Schuck-Ropke (THSR) wave function is formulated for the description of different structures of Be10. Physical properties such as energy spectrum and root-mean-square radii are calculated for the first two 0+ states and corresponding rotational bands. With only one single THSR wave function, the calculated results show good agreement with other models and experimental values. We apply, for the first time, the THSR wave function on the chain orbit (σ-orbit) structure in the 02+ state of Be10. The ring-orbit (π-orbit) and σ-orbit structures are further illustrated by calculating the density distribution of the valence neutrons. We also investigate the nonlocalized dynamics of α clusters and the correlations of two valence neutrons in Be10.

A microscopic calculation of $\ensuremath{\alpha}$-cluster preformation probability and $\ensuremath{\alpha}$-decay width in the typical $\ensuremath{\alpha}$ emitter $^{212}\mathrm{Po}$ is presented. Results are obtained by improving a recent approach to describe $\ensuremath{\alpha}$ preformation in $^{212}\mathrm{Po}$ [Phys. Rev. C 90, 034304 (2014)] implementing four-nucleon correlations (quartetting). Using the actually measured density distribution of the $^{208}\mathrm{Pb}$ core, the calculated $\ensuremath{\alpha}$-decay width of $^{212}\mathrm{Po}$ agrees fairly well with the measured one.

We explain how to treat a microscopic wave function of α-condensation taking a 3α-nucleus as a typical example. The wave function has been originally proposed ten years before by Horiuchi, Ropke, Schuck and the present author (Phys. Rev. Lett., 2001, 87: 192501). The microscopic model, which fully takes into account the Pauli principle between all the constituent nucleons, effective internucleon forces and the Coulomb force, can play an important role in reproducing an α-gas nature thanks to α-condensation as an excited state of α-like nuclei. An essential point of the wave function is to describe their ground state simultaneously. We study its typical features by giving an analytical formula of the norm kernel and the kernel concerning the one-body operator for 3α-condensation.

We present Skyrme-force Hartree–Fock calculations of metastable three-α chain-state configurations in 12 C , 16 C , and 20 C . A microscopic dynamical study of reactions involving three 4 He clusters shows that the triple-α linear chain configuration of 12 C is formed with a certain lifetime and subsequently decays into a triangular configuration of 12 C and then to a configuration near the ground state.

Strengths of monopole excitation to cluster states are discussed in light nuclei. They are in general so strong to be comparable to the single particle strength and shares an appreciable portion of the sum rule value, in spite of large difference of the structure between the cluster state and the shell-model-like ground state. We point out that the essential reasons of the large strengths are twofold: One is the fact that the clustering degree of freedom is possessed even in simple shell model wave functions (this fact is indicated by Bayman-Bohr theorem), and the other is due to the ground-state correlation induced by the activation of the cluster degree of freedom originally involved in the ground state. They are demonstrated in the case of 16 O as a typical light nucleus. The present results imply that the measurement of strong monopole transitions or excitations is generally very useful for the study of cluster states.

A nucleon is available for examining the property of α-condensate, which consists of composite bosons. Only a microscopic model, which fully takes into account the Pauli principle between all the constituent nucleons, can play its role. We give numerical results by equidistant spectrum model for the system of a neutron and 3α-cluster states. We discuss whether an extra nucleon can exist in the center of α-condensate or not.

Inelastic α scattering on 16 O is studied at 400 MeV using an ice target. Near the 4 α breakup threshold of 14.4 MeV, a broad peak is observed at an excitation energy of 13.6 ± 0.1 MeV with a width of 0.6 ± 0.2 MeV . The spin-parity is estimated to be 0 + from the momentum-transfer dependence. The observed width is significantly larger than that of the 0 + state at 12.0 MeV with the α + C 12 structure, indicating a well-developed α cluster structure. The magnitude of the cross section is sensitive to the density distribution of the constituent α clusters. The observed cross section is consistent with the theoretical prediction for the α cluster condensed state, characterized by its dilute density distribution with a large root-mean-square radius.

The onset of quartetting, i.e. α -particle condensation, in symmetric nuclear matter is studied with the help of an in-medium modified four nucleon equation. It is found that at very low density, quartetting wins over pairing because of the strong binding of the α particles. The critical temperature can reach values up to around 6 MeV. Also, the disappearance of α particles with increasing density, i.e. the Mott transition, is investigated. In finite nuclei the Hoyle state, that is, the 02+ of 12C is identified as an ‘ α -particle condensate’ state. It is conjectured that such states also exist in heavier n α nuclei, such as 16O, 20Ne, etc. Exploratory calculations are performed for the density dependence of the α -condensate fraction at zero temperature to address the suppression of the four-particle condensate below nuclear-matter density. Possible quartet condensation in other systems is discussed briefly.

A microscopic calculation of $\alpha$-cluster preformation probability and $\alpha$ decay width in the typical $\alpha$ emitter $^{212}$Po is presented. Results are obtained by improving a recent approach to describe $\alpha$ preformation in $^{212}$Po [Phys. Rev. C 90, 034304 (2014)] implementing four-nucleon correlations (quartetting). Using the actually measured density distribution of the $^{208}$ Pb core, the calculated alpha decay width of $^{212}$Po agrees fairly well with the measured one., Comment: 7 pages, 5 figures, 1 table, submitted to Phys. Rev. C

The nonlocalized aspect of clustering, which is a new concept for self-conjugate nuclei, is extended for the investigation of the $N\ensuremath{\ne}Z$ nucleus $^{9}\mathrm{Be}$. A modified version of the Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave function is introduced with a new phase factor. It is found that the constructed negative-parity THSR wave function is very suitable for describing the cluster states of $^{9}\mathrm{Be}$. Namely, the nonlocalized clustering is shown to prevail in $^{9}\mathrm{Be}$. The calculated binding energy and radius of $^{9}\mathrm{Be}$ are consistent with calculations in other models and with experimental values. The squared overlaps between the single THSR wave function and the Brink + generator coordinate method wave function for the $3/{2}^{\ensuremath{-}}$ rotational band of $^{9}\mathrm{Be}$ are found to be near 96%. Furthermore, by showing the density distribution of the ground state of $^{9}\mathrm{Be}$, the $\ensuremath{\pi}$-orbit structure is naturally reproduced by using this THSR wave function.

It is discussed that the recently observed 0+ state at 13.5 MeV in 16 O can be a good candidate of the 4α condensate. The 4α condensed state which is theoretically obtained in the vicinity of the 4α threshold is shown to have a relatively large α decay width, which is in well agreement with the value measured for the 0+ state at 13.5 MeV. The analysis of inelastic form factor of electron scattering to the 4α condensed state is made by artificially varying the size of the 4α condensate. Similarly to the case of Hoyle state, only the magnitude of the form factor is found to depend on its size, while the position of maximum and minimum are not changed.

New treatments are proposed for di-nucleon condensate based on the α condensate wave function. A beautiful mathematical structure is shown for the overlap.

In order to study non-zero spin excitations of the recently proposed alpha-cluster condensation in the self-conjugate 4n nuclei, spatial deformation is introduced into the model wave function of the alpha-cluster condensate. The rotational band states of 8Be are investigated as a first step of a test case for the study of the deformation of the alpha-cluster condensate. Calculations reproduce well the binding energy of the 0+ ground state and also the excitation energy of the 2+ state. Our 0+ wave function is found to be almost exactly equal to the 0+ wave function obtained by the generator coordinate method using Brink's 2 alpha wave function. The study shows that both the 0+ ground and 2+ excited states can be considered as having a gas-like (i.e. weakly bound) 2 alpha-cluster structure. As for the 0+ ground state, the change of the wave function, due to the introduction of the deformation, is found to be very small., Comment: 20 pages, 14 EPS figures

An effective $\ensuremath{\alpha}$-particle equation is derived for cases where an $\ensuremath{\alpha}$ particle is bound to a doubly magic nucleus. As an example, we consider $^{212}\mathrm{Po}$ with the $\ensuremath{\alpha}$ on top of the $^{208}\mathrm{Pb}$ core. We consider the core nucleus infinitely heavy, so that the $\ensuremath{\alpha}$ particle moves with respect to a fixed center; that is, recoil effects are neglected. The fully quantal solution of the problem is discussed. The approach is inspired by the Tohsaki-Horiuchi-Schuck-R\"opke wave function concept that has been successfully applied to light nuclei. Shell-model calculations are improved by including four-particle ($\ensuremath{\alpha}$-like) correlations that are of relevance when the matter density becomes low. In the region where the $\ensuremath{\alpha}$-like cluster penetrates the core nucleus, the intrinsic bound-state wave function transforms at a critical density into an unbound four-nucleon shell-model state. Exploratory calculations for $^{212}\mathrm{Po}$ are presented. Such preformed cluster states are very difficult to describe with shell-model calculations. Reasons for the different physics behavior of an $\ensuremath{\alpha}$-like cluster with respect to a deuteron-like cluster are discussed.

The theory of quartet condensation is further developed. The onset of quartetting in homgeneous fermionic matter is studied with the help of an in-medium modified four fermion equation. It is found that at very low density quartetting wins over pairing. At zero temperature, in analogy to pairing, a set of equations for the quartet order parameter is given. Contrary to pairing, quartetting only exists for strong coupling and breaks down for weak coupling. Reasons for this finding are detailed. In an application to nuclear matter, the critical temperature for alpha particle condensation can reach values up to around 8 MeV. The disappearance of alpha particles with increasing density, i.e. the Mott transition, is investigated. In finite nuclei the Hoyle state, that is the second 0+ state of 12C is identified as an 'alpha-particle condensate' state. It is conjectured that such states also exist in heavier n-alpha nuclei, like 16O, 20Ne, etc. The sixth 0+ state in 16O is proposed as an analogue to the Hoyle state. The Gross-Pitaevski equation is employed to make an estimate of the maximum number of alpha particles a condensate state can contain. Possible quartet condensation in other systems is discussed briefly., Contribution to MBT17 conference, Rostock, september 9-13, 2013, to be published in iop conference series

We present a new picture that the $\ensuremath{\alpha}$-linear-chain structure for $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$ has one-dimensional $\ensuremath{\alpha}$ condensate character. The wave functions of linear-chain states that are described by superposing a large number of Brink wave functions have extremely large overlaps of nearly 100% with single Tohsaki-Horiuchi-Schuck-R\"opke wave functions, which were proposed to describe the $\ensuremath{\alpha}$ condensed ``gaslike'' states. Although this new picture is different from the conventional idea of the spatial localization of $\ensuremath{\alpha}$ clusters, the density distributions are shown to have localized $\ensuremath{\alpha}$ clusters due to the inter-$\ensuremath{\alpha}$ Pauli repulsion.

The properties of α-cluster matter is studied within a microscopic framework, which takes full account of an effective inter-nucleon force andthe Pauli exclusion principle. The α-cluster matter plays an essential role in any α-cluster model as its infinite limit. This is because the α-cluster matter usedin an α-cluster model is under the same situation as is ordinary nuclear matter for an independent particle model. In this study, all the α-clusters are arranged in the generator coordinate space, which can be an equivalent model to the resonating group method. Every α-cluster is in the same situation if the infinite system has an ordered configuration like a crystal structure. In this article, simple lattice configuration is mainly considered from various aspects. In particular, distorted α-clusters are taken into account in order to see the properties of α-cluster in the infinite systems.

Gas-like α cluster states are investigated in 16O with the use of 4α OCM (orthogonality condition model) and GEM (Gauss expansion method). α-particle condensed state is found slightly above the 4α threshold as the \({0_6^+}\) state. The candidates of the \({\alpha + {^{12}{\rm C}(0_2^+)}}\) rotational states are also found at a few megaelectron volt above the condensate. The \({0_6^+}\) state is shown to have an analogue structure to the Hoyle state \({({^{12}{\rm C}}(0_2^+))}\) and its relation with the α+ Hoyle-like rotational states is also discussed.

Low density states near the 3α and 4α breakup thresholds in 12C and 16O, respectively, are discussed in terms of the α-particle condensation. Calculations are performed in Orthogonality Condition Model and Tohsaki–Horiuchi–Schuck–Ropke approaches. The \({0_2^+}\) state in 12C and the \({0_6^+}\) state in 16O are shown to have dilute density structures and give strong enhancement of the occupation of the S-state c.o.m. orbital of the α-particles. The possibility of the existence of α-particle condensed states in heavier nα nuclei is also discussed.

Gas-like α cluster states are investigated in 16O with the use of 4α OCM (Orthogonality Condition Model) and GEM (Gauss Expansion Method). α-particle condensed state is found slightly above the 4α threshold as the 06+ state. The candidates of the α+12C(02+) rotational states are also found at a few MeV above the condensate. The 06+ state is shown to have an analogue structure to the Hoyle state (12C(02+)) and its relation with the α+ Hoyle -like rotatinal states is also discussed. We further investigate the ground-state band in 20Ne by using the THSR-type wave function, which has been used for gas-like α cluster states. It is found that the single THSR wave function can describe well the corresponding states of the ground-state band in 20Ne, which indicates that even for describing non-gas-like cluster states the density parameter of the nucleus, which characterizes the THSR wave function, is more reasonable than the inter-cluster distance parameter, which characterizes the ordinary cluster model wave fun...

In this article we review the present status of alpha clustering in nuclear systems. First of all, an important aspect is condensation in nuclear matter. Second, the alpha container model, recently been proposed by Tohsaki-Horiuchi-Schuck-Roepke (THSR), will be outlined and the ensuing condensate aspect of the Hoyle state at 7.65 MeV in 12C will be investigated in some detail. After 15 years since the proposal of the alpha condensation concept a critical assessment of this idea will be given. Alpha gas states in other nuclei like 16O and 13C will be considered. The THSR wave function can also describe configurations of one alpha particle on top of a doubly magic core. The cases of 20Ne and 212Po will be investigated., review article

The stored ultra-cold neutrons have been developed. A high density ultra-cold neutron gas has been recently produced by using the nuclear spallation method. We investigate the thermodynamic properties of the quantum ultra-cold neutron gas in the Earth's gravitational field. We find that the quantum effects increase temperature dependence of the chemical potential and the internal energy in the low temperature region. The density distribution of quantum ultra-cold neutron gas is modified by the Earth's gravitational field., 7 pages, 4 figures, Submitted to Progress of Theoretical Physics

Isoscalar monopole strength function in ${}^{16}$O up to ${E}_{x}\ensuremath{\simeq}40$ MeV is discussed. We found that the fine structures at the low-energy region up to ${E}_{x}\ensuremath{\simeq}16$ MeV in the experimental monopole strength function obtained by the ${}^{16}$O$(\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}})$ reaction can be rather satisfactorily reproduced within the framework of the $4\ensuremath{\alpha}$ cluster model, while the gross three bump structures observed at the higher energy region ($16\ensuremath{\lesssim}{E}_{x}\ensuremath{\lesssim}40$ MeV) look likely to be approximately reconciled by the mean-field calculations such as random-phase approximation and quasirandom-phase approximation. In this paper, it is emphasized that two different types of monopole excitations exist in ${}^{16}$O; one is the monopole excitation to cluster states which is dominant in the lower-energy part (${E}_{x}\ensuremath{\lesssim}16$ MeV), and the other is the monopole excitation of the mean-field type such as one-particle, one-hole (1p1h), which is attributed mainly to the higher-energy part ($16\ensuremath{\lesssim}{E}_{x}\ensuremath{\lesssim}40$ MeV). It is found that this character of the monopole excitations originates from the fact that the ground state of ${}^{16}$O with the dominant doubly closed shell structure has a duality of the mean-field type as well as $\ensuremath{\alpha}$-clustering character. This dual nature of the ground state seems to be a common feature in light nuclei.

71 pages, 41 figures, review article, to be published in "Cluster in Nuclei (Lecture Notes in Physics) - Vol.2 -", ed. by C. Beck, (Springer-Verlag, Berlin, 2011); The $\alpha$-particle condensate in nuclei is a novel state described by a product state of $\alpha$'s, all with their c.o.m. in the lowest 0S orbit. We demonstrate that a typical $\alpha$-particle condensate is the Hoyle state ($E_{x}=7.65$ MeV, $0^+_2$ state in $^{12}$C), which plays a crucial role for the synthesis of $^{12}$C in the universe. The influence of antisymmentrization in the Hoyle state on the bosonic character of the $\alpha$ particle is discussed in detail. It is shown to be weak. The bosonic aspects in the Hoyle state, therefore, are predominant. It is conjectured that $\alpha$-particle condensate states also exist in heavier $n\alpha$ nuclei, like $^{16}$O, $^{20}$Ne, etc. For instance the $0^+_6$ state of $^{16}$O at $E_{x}=15.1$ MeV is identified from a theoretical analysis as being a strong candidate of a $4\alpha$ condensate. The calculated small width (34 keV) of $0^+_6$, consistent with data, lends credit to the existence of heavier Hoyle-analogue states. In non-self-conjugated nuclei such as $^{11}$B and $^{13}$C, we discuss candidates for the product states of clusters, composed of $\alpha$'s, triton's, and neutrons etc. The relationship of $\alpha$-particle condensation in finite nuclei to quartetting in symmetric nuclear matter is investigated with the help of an in-medium modified four-nucleon equation. A nonlinear order parameter equation for quartet condensation is derived and solved for $\alpha$ particle condensation in infinite nuclear matter. The strong qualitative difference with the pairing case is pointed out.

We propose a generalized wave function based on the flexible original THSR (Tohsaki, Horiuchi, Schuck, R\"opke) wave function [A. Tohsaki et al., Phys. Rev. Lett. 87, 192501 (2001)], which is applicable to studies of general cluster structures in nuclei. The ground-state band in ${}^{20}$Ne is investigated by using this generalized wave function and the energies obtained agree well with the experimental values. Moreover, it is found that the single generalized THSR wave functions almost completely coincide with the exact solutions of the $\ensuremath{\alpha}{+}^{16}$O resonating group method for the ground-state band in ${}^{20}\mathrm{Ne}$. For the ground state, for instance, the squared overlap between them is 99.3%. This indicates that the THSR model can also be extended to study more compact cluster states in nuclei such as, e.g., the ground-state band in ${}^{20}$Ne.

We present new effective internucleon forces responsible for the microscopic [alpha]-cluster model. We introduce the two- and three-body operators with finite ranges, which play an essential role in accounting for the bulk properties of nuclei in a broad range of masses including nuclear matter. The proposed forces are suitable for analyzing the elastic [alpha]-[alpha] scattering.

Nuclear matter in the generator coordinate space is studied introducing the tetrahedral α-clusters in α-cluster matter. Whether the phase transition from α-cluster matter to nuclear matter takes place as the density of nucleons increases is investigated using two kinds of effective inter-nucleon forces. Though the physical quantities strongly depend upon the properties of the effective internucleon force, the phase transition or the state close to the phase transition occurs in the nuclear interior. On the other hand, the surface never shows such a phase transition, and α-cluster matter is always more stable than nuclear matter. The calculated results thus indicate that α clusters float on the surface of nuclear matter

Low density states near the 3α and 4α breakup threshold in 12 C and 16 O , respectively, are discussed in terms of the α-particle condensation. Calculations are performed in OCM (Orthogonality Condition Model) and THSR (Tohsaki-Horiuchi-Schuck-Röpke) approaches. The [Formula: see text] state in 12 C and the [Formula: see text] state in 16 O are shown to have dilute density structures and give strong enhancement of the occupation of the S-state c.o.m. orbital of the α-particles. The possibility of the existence of α-particle condensed states in heavier nα nuclei is also discussed.

The α-particle condensed state in 16 O is investigated by the use of 4α OCM (Orthogonality Condition Model). Low-lying spectrum in experiment is well reproduced up to around the 4α break-up threshold. The excited state obtained around the 4α threshold as the [Formula: see text] state is shown to have a dilute density structure and give strong enhancement of the occupation of the S-state c.o.m. orbital of the α-particles. This is strong evidence that the state is regarded as the 4α condensed state.

The $4\ensuremath{\alpha}$ condensate state for $^{16}\mathrm{O}$ is discussed with the Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave function which has $\ensuremath{\alpha}$-particle condensate character. With a treatment of resonances, it is found that the $4\ensuremath{\alpha}$ THSR wave function yields a fourth ${0}^{+}$ state in the continuum above the $4\ensuremath{\alpha}$-breakup threshold, in addition to the three ${0}^{+}$ states obtained in a previous analysis. It is shown that this fourth ${0}^{+}$ $[({0}_{4}^{+}){}_{\mathrm{THSR}}]$ state has a structure analogous to that of the Hoyle state because it has a very dilute density and a large component of $\ensuremath{\alpha}+^{12}\mathrm{C}({0}_{2}^{+})$ configuration. Furthermore, single-$\ensuremath{\alpha}$ motions are extracted from the microscopic 16-nucleon wave function, and the condensate fraction and momentum distribution of $\ensuremath{\alpha}$ particles are quantitatively discussed. It is found that for the $({0}_{4}^{+}){}_{\mathrm{THSR}}$ state a large $\ensuremath{\alpha}$-particle occupation probability concentrates on a single-$\ensuremath{\alpha}$ $0S$ orbit and the $\ensuremath{\alpha}$-particle momentum distribution has a $\ensuremath{\delta}$-function-like peak at zero momentum, both indicating that the state has a strong $4\ensuremath{\alpha}$ condensate character. It is argued that the $({0}_{4}^{+}){}_{\mathrm{THSR}}$ state is the counterpart of the ${0}_{6}^{+}$ state which was obtained as the $4\ensuremath{\alpha}$ condensate state in the previous $4\ensuremath{\alpha}$ orthogonality condition model calculation and therefore is likely to correspond to the ${0}_{6}^{+}$ state observed at $15.1$ MeV. The necessity of including $\ensuremath{\alpha}+^{12}\mathrm{C}$ configurations in the THSR wave function is pointed out.

$\alpha$ particle condensation is a novel state in nuclear systems. We briefly review the present status on the study of $\alpha$ particle condensation and address the open problems in this research field: $\alpha$ particle condensation in heavier systems other than the Hoyle state, linear chain and $\alpha$ particle rings, Hoyle-analogue states with extra neutrons, $\alpha$ particle condensation related to astrophysics, etc., Comment: 12 pages. To be published in J. of Phys. G special issue on Open Problems in Nuclear Structure (OPeNST)