16 O + 16 O molecular structures of superdeformed states in S isotopes

. Structures of excited states in S isotopes are investigated by using the antisymmetrized molecular dynamics and generator coordinate method (GCM). The GCM basis wave functions are calculated via energy variation with a constraint on the quadrupole deformation parameter β . By applying the GCM after parity and angular momentum projections, the coexistence of positive- and negative-parity superdeformed (SD) bands are predicted in 33 - 36 S except for negative-parity states in 36 S. The SD bands have structures of 16 O + 16 O + valence neutron(s) in molecular orbitals around the two 16 O cores in a cluster picture. The conﬁgurations of the valence neutron(s) in the SD states are δ and/or π molecular orbitals. 34 4¯ conﬁgurations, same as of states in


Introduction
Clustering and deformation play important roles in nuclear structures. In S isotopes, 16 O + 16 O structures are expected to develop because S isotopes are analogous of Be isotopes. Be isotopes are considered to have structures of α + α + valence neutrons in molecular orbitals [1][2][3][4][5][6][7][8]. An α and an 16 O are double-closed-shell nuclei, and when α is replaced with 16 O in Be isotopes, they become S isotopes. In fact, existence of superdeformed (SD) states that contain a large amount of 16 O + 16 O cluster structure components has been predicted [9,10], which is an analogue of α + α structure of 8 Be. The dominant components of the SD states in 32 S are predicted to be 4hω configurations in the spherical shell-model picture.
By a γ-spectroscopy experiment, various positive-and negative-parity states have been observed [11]. Low-lying states are considered to be 0hω or 2hω neutron excited states. And many additional states are observed. But theoretical study about many-particle-many-hole states is insufficient.
In this paper, structures of positive-and negative-parity SD states in 34 S is discussed in details. The SD states have structures of 16 O + 16 O clusters and valence neutrons in molecular orbitals. Molecular structures developed in 33-36 S are also discussed. Details of this work is reported in Ref. [12].

The GCM and AMD models
In this work, wave functions of ground and excited states are calculated by using the antisymmetrized molecular dynamics (AMD) and the generator coordinate method (GCM), which is superposition of AMD wave functions to diagonalize Hamiltonian. The Gogny D1S force  Single-particle energies of as functions of quadrupole deformation parameter β for positive-(left) and negative-parity (right) states. Circles and squares show positive-and negative-parity orbits, respectively. Numbers in brackets show the Nilsson quanta for the two highest orbits of neutrons in SD region. This figure is taken from Ref. [12].
is used as an effective Hamiltonian. An AMD wave function is a Slater determinant of Gaussian wave packets, and parameters in the wave functions such as position of each wave packet and spin directions are optimized by variational calculation with a constraint on quadrupole deformation parameter β. Obtained wave functions are projected onto eigen states of parity and angular momentum and superposed. Final wave functions are obtained by diagonalizing Hamiltonian. Figure 1 shows β-energy surface obtained by energy variational calculations for positive-and negative-parity states, respectively, in 34 S. β-energy surfaces have local minima and a shoulder at SD region around β ∼ 0.6, which implies coexistence of positive-and negative-parity SD states. Figure 2 shows single-particle orbits of neutrons as functions of quadrupole deformation parameter β for positive-and negative-parity states, respectively. Circles and squares show positive-and negative-parity orbits, respectively. In SD region, the two highest orbits are flat exp. theory Figure 3. Left and right parts show experimental and theoretical level scheme, respectively, in 34 S. This figure is taken from Ref. [12]. and those Nilsson quanta are [202] and [321], which are positive-and negative-parity orbits, respectively. For lower 16 orbits, they have 2hω excited configurations. Proton orbits have same configurations as lower 16 orbits of neutrons. Totally, the 32 S core part of SD region in 34 S has 4hω excited configurations, which are same as configurations of predicted SD states in 32 S [ 32 S(SD)] [10]. The 32 S(SD) core part has neck structure as well as a prediction of SD states. They show that structure of these wave functions are 32 S(SD) core + [202] and [321] neutrons.

Results
Superposing those wave functions after parity and angular momentum projection, level scheme is obtained as shown in Fig. 3. Various rotational band are obtained. Three bands labeled K π = 0 + SD1 , 4 − SD , and 0 + SD2 , coexist, which are SD bands. Those three SD bands have multiparticle-multi-hole excited configurations for both of proton and neutron parts. Configurations of dominant components of K π = 0 + SD1 , 4 − SD , and 0 + SD2 bands are 32 S(SD) core +  Fig. 4. The left part of Fig. 4 shows schematic pictures of phase of 0d orbits around two 16 O cores (dotted circles) for |l z | = 2, 1, and 0 components. By linear combination of 0d around two 16 O cores, δ, π, and σ orbitals (right part) are generated from |l z | = 2, 1, and 0 orbits, respectively, around two 16 O cores. Figure 4(a) shows a |l z | = 2 orbital, which is a δ orbital. Superposing two |l z | = 2 orbits around 16 O cores, a molecular orbital becomes right part. It has no node for z-direction. The 0d orbits has no node for radial direction, so Nilsson quanta of a δ orbital is [202]. Figure 4(b) shows a |l z | = 1  Figure 4. Schematic illustrations of molecular orbitals generated from 0d orbits around two 16 O cores for (a) δ, (b) π, and (c) σ orbitals. Horizontal axis is z-axis. Inverse triangles show locations of nodes in molecular orbitals in the z direction. Numbers in brackets show Nilsson quanta. This figure is taken from Ref. [12].
orbital, which is a π orbital. As same discussions, The π orbital have two nodes for z-directions, and Nilsson quanta of a π orbital is [321]. Figure 4(c) shows a |l z | = 0 orbital, which is a σ orbital. Nilsson quanta of a σ orbital is [400]. Totally, structures of K π = 0 + SD1 , 4 − SD , and 0 + SD2 are interpreted as 16 O + 16 O + δ 2 , δπ, and π 2 structures, respectively, in a cluster picture. It is predicted that 33-36 S also have positive-and negative-parity SD states with 16 O + 16 O molecular structure except for negative-parity states in 36 S.

Conclusions
In conclusions, structures of SD states in 34 S by using the AMD and the GCM. In 34 S, two positive-parity and one negative-parity SD bands coexist. Those structures are interpreted as 16 O + 16 O + δ 2 , π 2 and δπ molecular orbitals around two 16 O cores. 33-36 S also have SD states except for negative-parity states in 36 S. They have also molecular structures. In order to understand largely deformed states in N = Z nuclei, clustering and molecular orbital around the cluster core are important.