Application of the generator coordinate method to neutron-rich Se and Ge isotopes

The quantum-number projected generator coordinate method (GCM) is applied to the neutron-rich Se and Ge isotopes, where the monopole and quadrupole pairing plus quadrupole-quadrupole interaction is employed as an e ffective interaction. The energy spectra obtained by the GCM are compared to both the shell model results and the experimental data. The GCM reproduces well the energy levels of high-spin states as well as the low-lying states. The structure of the low-lying collective states is analyzed through the GCM wave functions.


Introduction
The intriguing properties of the even-even Se and Ge isotopes in the mass region A ∼ 80 have been investigated in a number of previous experimental and theoretical studies [1][2][3].These isotopes belong to a typical transitional region that lies between spherical and deformed regions.The structure of their low-lying states can be attributed to the interplay of rotational and vibrational collective motions.For high-spin states, γ-ray spectroscopy of the near-yrast states in the N = 44 and 46 isotones of Se ( 80,82 Se) was carried out for deep-inelastic reactions [4].
Recently, full-fledged shell-model calculations were performed on the even-even and odd-mass nuclei in this mass region [5].The calculations reproduced well the experimental energy levels and electromagnetic transition rates for both the low-lying and high-spin states.In order to investigate the features of these states, the energy spectra in the shell model were compared with those in a pair-truncated shell model (PTSM).In the model, the full shell model space is truncated within the subspace where the collective nucleon pairs with various angular momenta are assumed to be its building blocks.The same set of the interactions as employed in the shell model calculations was applied to even-even nuclei, and an excellent agreement with the shell model result was achieved for energy levels.Through the analysis of the PTSM wave functions, it was found that the angular momenta zero and two collective pairs are dominant in low-lying states, while the effect of the alignment of two 0g 9/2 neutrons becomes apparent above 8 + 1 states.In the present study, we apply the quantum-number-projected generator coordinate method (GCM) to 78 Ge under the same interaction as used in the previous shell model study [5].We do not discuss other nuclei, but similar results are obtained for 76 Ge, 80 Se, and 78 Se.Their results and the detailed prescriptions for the GCM will be presented in a forthcoming paper [6].

Theoretical framework
The previous GCM studies [7,8] made it clear that, for a description of the nuclear collective and single-particle motions in a transitional region, the angular momenta of the neutron and proton systems (I ν and I π ) should be projected out separately, and the total spin I is constructed by angular momentum coupling.Thus in the present GCM calculations, the angular momentum projection is performed separately in each proton or neutron space.To generate functions for the GCM, we employ the intrinsic states Φ τ (β, γ) ⟩ for either neutron or proton system (τ = ν or π), where β and γ indicate axial and triaxial quadrupole deformations, respectively.The ρth GCM wave function with angular momentum I τ in either neutron or proton space is given by where PI τ M τ K τ is the spin projection operator, F I τ i K τ ρ represents the weight function to be determined by solving the Hill-Wheeler equation, and i stands for a representative point with deformation (β, γ).Then, the many-body wave function for an even-even nucleus can be written as where I is the total spin and M is its projection.The intrinsic state Φ τ (β, γ) ⟩ is constructed by the following procedure.First we consider the intrinsic Nilsson hamiltonian for either neutron or proton space: The intrinsic single-particle deformed state ατ by diagonalizing the Nilsson hamiltonian ĥτ .In order to include the pairing correlation, we introduce the nucleon pair creation operator as where ᾱ indicates the time reversal state of α.Then the intrinsic many-body states for neutrons or protons are written as where N denotes the normalization.Then, the structure coefficients f 's are determined by variation for a definite deformation (β, γ): Here, Ĥτ is the spherical Hamiltonian between like nucleons (τ = ν or π).The potential energy surface (PES) is defined as
The ground state energies for the 78 Ge are E SM = −11.0530MeV, E GCM(triaxial) = −11.0428MeV, and E GCM(axial) = −10.9643MeV, respectively for the SM, the GCM with triaxial deformations and the GCM only with axial deformations.A large improvement is seen assuming the triaxial shape for the ground state energy.

Figure 1 .
Figure 1.The contour plot of the PES calculated for neutrons (left panel) and protons (right panel) in 78 Ge.The contour line separation is 0.05 MeV.