Ground state of nuclei in the Barium region within the Highly Truncated Diagonalization Approach

The ground state of some barium isotopes has been investigated in the framework of the paritysymmetry projection of the Highly Truncated Diagonalization Approach (HTDA), which is suited to treat the correlations in an explicitly particle-number conserving microscopic approach. A Skyrme energydensity function using the SkM∗ interactions has been considered to treat the particle-hole channel, whereas a density-independent δ force has been adopted for the residual interaction. The obtained results are compared with the previous calculations using the Woods-Saxon potentials.


Introduction
It is well known that the nuclear structure of doubly-closed shell nuclei is well described by the mean-field approximations. However, many correlations (e.g. pairing correlations) must be taken into account when the nuclear system is far from the stability on the nuclear chart. One simple and useful way to deal with this problem is to use the Bardeen-Cooper-Schrieffer (BCS) approximation. The latter was firstly designed for the system of about 10 23 fermions in solid-states physics. Unfortunately, this approximation breaks the particle-number symmetry in the nuclear finite systems. The Highly Truncated Diagonalization Approach (HTDA) has been developed to overcome this difficulty [1]. This approximation produces the realistic correlated wave functions in an approach which explicitly conserves the particle number and does not violate the Pauli principle. It consists in performing intrinsic shell-model like calculations using single-particle states deduced from a mean-field potential. The nucleon-nucleon effective interactions (Skyrme-type interactions [2]) have been used to define the mean field while the zero-range δ force has been used to simulate the residual interactions. Recently a parity-symmetry projection of correlated microscopic solutions for HTDA approach has been extensively used to study the octupole deformation for 240 Pu (at second fission barrier region) [3] and 194 Pb (normal deformed and superdeformed states) [4] where the strong octupole correlations can arise when nucleons near the Fermi surface occupy states of opposite parity with orbital and total angular momenta differing by 3ħ.
In this work, we extend the latter calculations for the barium isotopes region which have been successfully performed by using the semi-microscopic approach [5]. The paper is organized as follows. In Sec. 2, we present the formalism of the parity-symmetry projection for the HTDA solutions. The analysis of the numerical results is presented in Sec. 3. The main conclusions of our study are drawn in the last section.

Theoretical formalism
The HTDA approach solves the Schrӧdinger equation in a highly truncated space. This space contains the excitations beyond 1p-1h excitations. Hence, the HTDA wave function is a superposition of Slater determinants built from np-nh excitations on the ground-state obtained from a converged Hartree-Fock+BCS (HF+BCS) calculations.
The Hamiltonian of the many-body nuclear system reads We define the independent quasiparticles Hamiltonian and † The HTDA Hamiltonian could be re-written as follows The total energy HTDA E is defined as the average value of Ĥ for the correlated state where  is the eigenvector of lowest energy. It conserves the particle-number symmetry where the real coefficient i  ensures the normalization of the correlated state Finally, the projected energy reads where     ,  is the parity operator and p = ±1.

Numerical aspects
The numerical procedure could be divided into 3 steps. The parity-asymmetry HF + BCS calculations have been performed to obtain the ground state and the mean-field potential. The effective phenomenological Skyrme SkM * interaction has been chosen. The intensity of the seniority force is fixed for 148 Ba and for the remaining isotopes. The strengths gq (where q is a charge state index) of the seniority force are Then, a projection after variation (PAV) has been performed to get the parity-projected HTDA ground state energy at a fixed quadrupole and octupole deformation.
Numerical integrations have made use of a Gauss-Laguerre and Gauss-Hermite approximations scheme with 16 and 50 mesh points, respectively.

Results
The HTDA solutions have been projected on the good parity state for each value of octupole moments 30 Q . The obtained results (see Fig. 1) show that the positive-parity curve is always below the un-projected HTDA curve whereas the negative-parity curve is always above it. At zero octupole deformation, the un-projected HTDA solution and the projection on positive-parity states have the same energy. The projection energy on the negative-parity state is not defined. At large octupole moment, the three curves merge. In contrast, if the symmetrical equilibrium solution has a sufficiently small deformation, the parity-projection could create a minimum at a non-vanishing value of the octupole moment. This situation was also observed in the calculations of intrinsic super-deformed state of 194 Pb [4]. A static instability away from symmetry, (which was not available with the intrinsic solutions), appears. For the situation where the intrinsic equilibrium solution is not symmetrical but corresponds to a not-too-large absolute value of 30 |Q | , the positive-parity projection emphasizes the instability already present at the intrinsic level, yielding possible the corresponding equilibrium 30 |Q | values. Beyond a certain critical value of 30 |Q | , the projection does not have any effects.

Conclusion
We have studied the octupole deformation properties of nuclei in the barium region by using the microscopic self-consistent HTDA approach. Four even-even neutron rich nuclei are considered. The obtained results confirm the calculations of W. Nazarewicz et al. which predicted a significant octupole deformation in the barium region. Note that the latter calculations were performed by using the Woods-Saxon potentials.