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.

The α-bosonic properties such as single-α orbits and occupation numbers in J π =0 + , 2 + , 1 − and 3 − states of 12 C around the 3α threshold are investigated with the semi-microscopic 3α cluster model. As in other studies, we found that the 0 + 2 (2 + 2 ) state has dilute-3α-condensate-like structure in which the α particle is occupied in the single S (D) orbit with about 70 % (80 %) probability. The radial behaviors of the single-α orbits as well as the occupation numbers are discussed in detail in comparison with those for the 0 + 1 and 2 + 1 states together with the 1 − 1 and 3 − 1 states.PACS numbers: 21.10.Dr, 21.10.Gv, 21.60.Gx, 03.75.Hh (B) J (a) is totally symmetric for any two-α-cluster exchange. Thus, it has bosonic property. From the RGM equation (8), Φ (B) J should satisfy the following equation N −1/2 HN −1/2 − E Φ (B) J

Energy levels of the double-Λ hypernuclei Λ 7 Λ He, Λ 7 Λ Li, Λ 8 Λ Li, Λ 9 Λ Li, Λ 9 Λ Be and Λ 10 Λ Be are predicted

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.

Movitated by the on-going gamma-ray experiments at the Alternating Gradient Synchrotron at Brookhaven National Laboratory, we discuss the energy splittings of the 5/2(+)(1)-3/2(+)(1) doublet in (9)(Lambda)Be and the 3/2(-)(1)-1/2(-)(1) doublet in (13)(Lambda)C for which the LambdaN spin-orbit ( LS) and antisymmetric spin-orbit ( ALS) forces are relevant. In the microscopic 2alpha+Lambda ( 3alpha+Lambda) model for (9)(Lambda)Be ( (13)(Lambda)C), all the available Nijmegen one-boson-exchange (OBE) model LambdaN interactions lead to a wide range of splittings of 0.08-0.20 MeV in (9)(Lambda)Be and 0.39-0.96 MeV in (13)(Lambda)C. On the other hand, if we use information from quark-model LambdaN interactions which have generally large ALS force, the splittings become about half of the smallest OBE model prediction.

Dilute multi α cluster condensed states with spherical and axially deformed shapes are studied with the Gross-Pitaevskii equation and Hill-Wheeler equation, where the α cluster is treated as a structureless boson. Applications to self-conjugate 4N nuclei show that the dilute N α states of 12 C to 40 Ca with J π = 0 + appear in the energy region from threshold up to about 20 MeV, and the critical number of α bosons that the dilute N α system can sustain as a self-bound nucleus is estimated roughly to be N cr ∼ 10. We discuss the characteristics of the dilute N α states with emphasis on the N dependence of their energies and rms radii.

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].

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