Unified description of shell and cluster coexistence in 16 O with a five-body model

A famous and long-standing problem so-called the “mysterious” 0 state in 16O is discussed with a single scheme, a 12C plus four-nucleon five-body model. The wave function is expressed in terms of explicitly correlated Gaussian functions. Preliminary results show that the energy levels are in good agreement with experiment. A welldeveloped 12C+α cluster structure appears in the first excited 0 state, whereas the ground state has shell model like structure.


Introduction
A cluster structure is one of the most interesting phenomena in the nuclear many-body system and often appears in the excited spectrum around doubly closed shell nuclei.A long-standing problem so-called the "mysterious" 0 + state in 16 O is a famous example of exhibiting both cluster and shell structure in its spectrum.The ground state of 16 O has been recognized as having a doubly closed shell model configuration, and thus the first excited state is expected to have the negative parity state in accordance with the particle-hole excitation but actually observed as 0 + state.Many theoretical works are devoted to understand the mysterious state, however, the state has not been reproduced microscopically even in modern large scale calculations [1].Only 12 C+α cluster model succeeds to describe both the ground and excited states with 0 + as well as the parity inverted partner of these [2].It is quite challenging problem to describe such systems in a single scheme, however, the unified description is desired for its deep understanding.Since a fully microscopic calculation is not feasible at present, it is interesting to study 16 O with a 12 C core plus four-nucleon model, that is without assuming the alpha cluster.
In this contribution, we present our attempt at describing such coexistence with the 12 C+fournucleon five-body model.In the next section, we briefly explain our approach and present preliminary results for 16 O as well as 16 C in Section 3. Summary and future prospects are drawn in Section 4.

Method: A core plus four-nucleon model
A five-body system we consider here is characterized by the Hamiltonian which consists of a nucleonnucleon (N − N) potential for valence nucleons and a 12 C-nucleon (C−N) potential.A central Minnesota (MN) potential [3] is employed as the N − N potential.Symmetrized Woods-Saxon and its a e-mail: whoriuchi@nucl.sci.hokudai.ac.jp derivative forms are assumed for the C−N potential, and their parameters are determined to reproduce the low-lying spectrum of 13 C with 1/2 − , 1/2 + , and 5/2 + .Our C−N potential is deep enough to accommodate some redundant or Pauli-forbidden states.To eliminate such states we impose the orthogonality constraint for the relative motion of the valence nucleons, which is practically achieved by adding a pseudo potential to the Hamiltonian [4].Here we assume the harmonic oscillator wave functions of 0s 1/2 and 0p 3/2 as the occupied orbits of 12 C.We express the five-body wave function with total angular momentum J in terms of a linear combination of many basis states of the following form where A is an antisymmetrizer, Y L (r) is a solid spherical harmonic.The χ S M S (η T M T ) is the spin (isospin) function of the valence nucleons.The 4 × 4 matrix A and 4-dimensional vectors u 1 and u 2 are variational parameters to be optimized and a tilde denotes a transpose of a matrix.It is noted that all coordinates are explicitly correlated, which enables us to obtain a precise solution of a many-body Schrödinger equation [5].An advantage of this method is that its functional form does not change under any coordinate transformation, and thus both cluster-and shell-model like configurations can be expressed in a single scheme.Because of this flexibility the method have been applied to many quantum mechanical systems (See recent review [6]).In particular, the application to the spectrum of 4 He is interesting as an analog of the 16 O spectrum.Similarly to 16 O the first excited state of 4 He has J π = 0 + that is shown to have a cluster state of 3N + N ( 3 H+p and 3 He+n) configuration [7,8].
Since we have to determine a large number of the variational parameters an efficient algorithm is needed to get the converged solution with limited computational resources.One of the methods is the stochastic variational method [9,10].Assuming that we already have some basis states, parameters of the next basis are determined by the following procedure: First we generate several candidates with randomly chosen parameters and select the basis which gives the lowest energy among them, and then include it to the set of bases.We repeat this procedure until the energy is converged.

Preliminary results and discussion
Figure 1 displays the calculated energy curves of 16 C and the ground and first excited states of 16 O as functions of the number of basis functions.The energy of 16 C is converged at 7000 basis states and the energy appears very closely to the experiment.The energy curves of 16 O are drawn in the right panel and exhibit very slow convergence compared to that of 16 C due to the difficulty of eliminating the forbidden states.Figure 2 plots the components of the forbidden states in the A = 16 wave functions.The components gradually decrease as increasing the basis dimension.The values are around 10 −4 at 4000 basis dimension which corresponds to the ground state energy of 16 O crosses the 12 C+α threshold.Most of basis states are used for removing the forbidden states.The case of 16 C is easier than that of 16 O because the valence neutrons do not form a strongly bound cluster.This is because, when the valence nucleons form a cluster, the basis states may have a large overlap with the forbidden states, and thus more basis states are needed to get a converged solution.In fact, the components of the forbidden states decreases monotonically beyond 4000 basis in the case of 16 C, whereas the two 0 + states in 16 O show some oscillatory behaviors.However, the preliminary results for 16 O are promising.The ground state appears below the 12 C+α threshold and the first excited state appears just at a few MeV from the ground state, consistently with experiment.The calculation is still being in progress.
It is interesting to see how the cluster structure develops with looking at the expectation values of some Hamiltonian pieces.Figure 3  acting among the valence nucleons as a function of the number of basis dimensions.For 16 C, the contribution of the C−N potential is very large compared to the N − N contribution, and thus the state can be regarded as having a shell-model like state in which the valence nucleons move almost independently around the core.Reflecting above simple structure, two energy curves of 16 C shows a constant behavior with the increasing number of basis.In contrast to this, the curves for the ground state of 16 O changes drastically at 4000 to 6000 basis states where the energy crosses over the 12 C+α threshold (See Fig. 1).We expect the structure change from the cluster structure to shell-model like structure at this basis dimension.Beyond 8000 basis states where the energies are close to the convergence, the C−N contribution obtained is similar to that of 16 C, whereas the N − N contribution is larger.The N − N energy is found approximately a half of the N − N contribution of the free alpha particle (−88.03MeV) obtained with the MN potential [5].In the ground state of 16 O, the α particle is strongly distorted by attraction of the core nucleus as well as influence of the forbidden states.In the case of the first excited state of 16 O, the N − N contribution is dominant and is very close to that of the free alpha particle.This implies that the first excited state has a well developed 12 C+α structure as predicted by the cluster model [2].It should be noted that the expectation value of the C−N potential in the ground states of 16

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences,

Figure 1 .Figure 2 .
Figure1displays the calculated energy curves of16 C and the ground and first excited states of16 O as functions of the number of basis functions.The energy of16 C is converged at 7000 basis states and the energy appears very closely to the experiment.The energy curves of16 O are drawn in the right panel and exhibit very slow convergence compared to that of 16 C due to the difficulty of eliminating the forbidden states.Figure2plots the components of the forbidden states in the A = 16 wave functions.The components gradually decrease as increasing the basis dimension.The values are around 10 −4 at 4000 basis dimension which corresponds to the ground state energy of16 O crosses the 12 C+α threshold.Most of basis states are used for removing the forbidden states.The case of16 C is easier than that of16 O because the valence neutrons do not form a strongly bound cluster.This is because, when the valence nucleons form a cluster, the basis states may have a large overlap with the forbidden states, and thus more basis states are needed to get a converged solution.In fact, the components of the forbidden states decreases monotonically beyond 4000 basis in the case of 16 C, whereas the two 0 + states in16 O show some oscillatory behaviors.However, the preliminary results for16 O are promising.The ground state appears below the 12 C+α threshold and the first excited state appears just at a few MeV from the ground state, consistently with experiment.The calculation is still being in progress.It is interesting to see how the cluster structure develops with looking at the expectation values of some Hamiltonian pieces.Figure3displays the expectation values of C−N and N − N potentials

Figure 3 .
Figure 3. Expectation values of C−N and N − N potentials for the ground state of 16 C, the ground state of 16 O (0 + 1 ), and the first excited state of 16 O (0 + 2 ) from left to right, respectively, as functions of the number of basis.