A new α-cluster wave function is proposed which is of the α-particle condensate type. Applications to 12 C and 16 O show that states of low density close to the 3 resp. 4 α-particle threshold in both nuclei are possibly of this kind. It is conjectured that all self-conjugate 4n nuclei may show similar features. Keywords: Binding energies, Collective levels, α-like correlations, Bose-Einstein condensation, nuclear matter, α-particle matter. PACS: 03.75.F, 21.10. Dr, 21.10.Re, 21.65.+f Many properties of finite quantum systems such as nuclei, quantum dots, atomic clusters or ultracould gases in a trap are fairly well described within a mean-field approximation neglecting the correlations between the quasiparticles. However, sometimes correlations become strong, giving rise to the formation of clusters, and they have to be taken into account. An intriguing problem is when bosonic clusters as bound states of fermions are produced, and the Bose character of the composite clusters competes with the fermionic properties of their constituents. As an example, we will discuss the relevance of α-like four-nucleon correlations in atomic nuclei. Special attention will be paid to such correlations which correspond to an α-type condensate in low-density symmetric nuclear matter, similar to the Bose-Einstein condensation observed for finite numbers of bosonic atoms such as Rb or Na in traps.It is a well known fact that in light nuclei many states are of the cluster type [1][2][3][4]. In the case of cluster states of stable nuclei where we have only very few excess nucleons in addition to the clusters, they are all located close to or above the threshold energy of breakup into constituent clusters. This fact which is known as the threshold rule [5] means that the inter-cluster binding is weak in cluster states. The threshold rule can be considered as a necessary condition for the formation of the cluster structure, because if the inter-cluster binding is strong the clusters overlap strongly and the clusters will loose their identities.For example one of the fundamental questions of the cluster model is what kind of α-particle cluster states can be expected to exist around the threshold energy E thr nα = nE α of nα breakup in self-conjugate 4n nuclei. One possible answer to this question, which is strongly under debate, is the existence of the cluster state of a linear nα chain structure. The idea of the linear α chain state, originally due to Morinaga [6], is so fascinating that recently the formation of linear 6α chain states in 24 Mg was studied extensively by experiments and also theoretically [7]. The possibility of the linear 3α chain state in 12 C, which is the simplest linear α chain state was studied in detail by many authors solving the 3α problem microscopically [4]. However, these three-body studies all showed that the 3α-cluster states around the 3α threshold energy E thr 3α do not have a linear chain structure. For example it was found that the calculated second 0 + state in 12 C, which corresponds to the observe...

The wave function of the second 0 + state of 12 C which was obtained long time ago by solving the microscopic 3α problem is shown to be almost completely equivalent to the wave function of the 3α condensed state which has been proposed recently by the present authors. This equivalence of the wave functions is shown to hold in two cases where different effective two-nucleon forces are adopted. This finding gives strong support for interpreting the second 0 + state of 12 C which is the key state for the synthesis of 12 C in stars ( 'Hoyle' state ), and which is one of the typical mysterious 0 + states in light nuclei, as a gas-like structure of three α particles, Bose-condensed into an identical s-wave function.The α clustering nature of the nucleus 12 C has been studied by many authors using various approaches [1]. Among these studies, solving the fully microscopic threebody problem of α clusters gives us the most important and reliable theoretical information of α clustering in 12 C within the assumption that no α cluster is distorted or broken except for the change of the size parameter of the α cluster's internal wave function. As representatives for the solution of the microscopic 3α problem where the antisymmetrization of nucleons is exactly treated, we here quote two works: one by Uegaki et al.[2] and the other by Kamimura et al. [3] both of which were published almost a quarter century ago. In these works, the 12 C levels are described by the wave function of the form A{χ(s, t)φ 3 α } with A standing for the antisymmetrizer, φ 3 α ≡ φ(α 1 )φ(α 2 )φ(α 3 ) for the product of the internal wave functions of 3 α clusters, and s and t for the Jacobi coordinates of the center-of-mass motion of 3 α clusters. Here φ(α i ) (i = 1, 2, 3) is the internal wave function of the α-cluster α i having the form φ(The wave function χ(s, t) of the relative motion of 3 α clusters is obtained by solving the eigen-energy problem of the full three-body equation of motion; φ Recently, based on the investigations by Röpke, Schuck, and coauthors on the possibility of α-particle condensation in low-density nuclear matter [6], the present authors proposed a conjecture that near the nα threshold in self-conjugate 4n nuclei there exist excited states of dilute density which are composed of a weekly

Certain aspects of the recently proposed antisymmetrised α particle product state wave function, or THSR α cluster wave function, for the description of the ground state in 8 Be, the Hoyle state in 12 C, and analogous states in heavier nuclei, are elaborated in detail. For instance, the influence of antisymmetrisation in the Hoyle state on the bosonic character of the α particles is studied carefully. It is shown to be weak. Bosonic aspects in Hoyle and similar states in other self-conjugate nuclei are, therefore, predominant. Other issues are the de Broglie wave length of α particles in the Hoyle state which is shown to be much larger than the inter-alpha distance. It is pointed out that the bosonic features of low density α gas states have measurable consequences, one of which, that is enhanced multi-alpha decay properties, likely already have been detected. Consistent with experiment, the width of the proposed analogue to the Hoyle state in 16 O at the excitation energy of Ex = 15.1 MeV is estimated to be very small (34 keV), lending credit to the existence of heavier Hoyle-like states. The intrinsic single boson density matrix of a self-bound Bose system can, under physically desirable boundary conditions, be defined unambiguously. One eigenvalue then separates out, being close to the number of α's in the system. Differences between Brink and THSR α cluster wave functions are worked out. No cluster model of the Brink type can describe the Hoyle state with a single configuration. On the contrary, many superpositions of the Brink type are necessary, implying delocalisation towards an α product state. It is shown that single α particle orbits in condensates of different nuclei are almost the same. It is thus argued that α particle (quartet) antisymmetrised product states of the THSR type are a very promising novel and useful concept in nuclear physics.

To explore the four-alpha-particle condensate state in 16O, we solve a full four-body equation of motion based on the four-alpha-particle orthogonality condition model in a large four-alpha-particle model space spanned by Gaussian basis functions. A full spectrum up to the 0_{6};{+} state is reproduced consistently with the lowest six 0;{+} states of the experimental spectrum. The 0_{6};{+} state is obtained at about 2 MeV above the four-alpha-particle breakup threshold and has a dilute density structure, with a radius of about 5 fm. The state has an appreciably large alpha condensate fraction of 61%, and a large component of alpha+12C(0_{2};{+}) configuration, both features being reliable evidence for this state to be of four-alpha-particle condensate nature.

We adopt a personal approach here reviewing several calculations over the years in which we have experienced confrontations between cluster models and the shell model. In previous cluster conferences we have noted that cluster models go hand in hand with Skyrme Hartee-Fock calculations in describing states which cannot easily, if at all, be handled by the shell model. These are the highly deformed (many particle-many hole) intruder states, linear chain states e.t.c. In the present work we will consider several topics; the quadrupole moment of 6 Li, the non-existence of low lying intruders in 8 Be, and then jumping to the f 7/2 shell, we discuss the two-faceted nature of the nuclei-sometimes displaying shell model properties, other times cluster properties. I. THE QUADRUPOLE MOMENT OF THE J = 1 + STATE IN 6 LI Whereas the quadrupole moment of the deuteron is positive(Q = +2.74mb), that of the J=1 + state of 6 Li is negative, Q=-0.818(17)mb. The magnetic moment of the deuteron is µ = 0.85741 nm while that of 6 Li is 0.822 nm. There appears to be a big discrepancy between cluster model calculations and the shell model calculations. In nearly all cluster model calculations Q comes out positive. However in many shell model calculations Q comes out negative, sometimes too negative. This is an important problem that deserves further attention. See for example arguments in the literature between the cluster group[1] and the shell model group[2]. See also the recent compendium of A=6 by D.R. Tilley et. al.[3].