Potential-energy-surface calculations.

The discussion of non-spherical nuclear shapes is essential in order to understand the nuclear behaviour of $\gamma$-deformed and $\gamma$-soft nuclei. For the purpose of calculating the nuclear shape of specific multi-quasiparticle configurations, the Configuration-Constrained Blocking Method (CCBM), developed by Xu et al. [17] has been chosen on account of its successful predictions for the region of interest [18]. Unlike other theoretical calculations which are limited to axially symmetric shapes [19], the CCBM includes the axially asymmetric $\gamma$-degree of freedom. This is essential in order to get realistic theoretical predictions for this mass region since the $\gamma$-deformation and $\gamma$-softness of a nucleus leads to configuration mixing and alters the decay properties of the states. Other theoretical models such as the rigid triaxial rotor model (RTRM) [20] include the $\gamma$-degree of freedom, but unlike the CCBM, the RTRM describes the nuclear shape with constant values of the deformation parameters, $\beta$ and $\gamma$. This is in contrast to the CCBM, where $\beta$ and $\gamma$ are treated as dynamical variables. The calculations used in this work employ the non-axial deformed Woods-Saxon potential [7] with universal parameters [21]. The Lipkin-Nogami description is employed to avoid the sudden collapse of proton pairing around closed shells [11]. In this method of shape calculation, the pairing strength, G, is determined by the gap method [10]. The precise value of G is of particular importance in the calculation of excitation energies of multi-quasiparticle configurations, but does not have a large effect on the determination of shapes [18].

According to the Strutinsky theorem [22] the total energy of a nucleus can be decomposed into a macroscopic and a microscopic part given by,

$$E_{total}(\beta_2,\gamma,\beta_4)=E_{macroscopic}+\delta E_{microscopic}$$ (20)

where the macroscopic part is obtained from the standard liquid drop model [23] and the microscopic part is calculated within the Woods-Saxon model [7]. $\delta$E $_{microscopic}$ can be expressed as $\delta$E $_{microscopic}$=E$_{LN}$-E $_{shell -correction}$, where E$_{LN}$ is the pairing energy obtained using the Lipkin-Nogami (LN) treatment of pairing [11] and E $_{shell -correction}$ is the Strutinsky shell-correction [22]. The configuration energy in the LN approach is given by,

$$E_{LN} = \sum_{j=1}^S \epsilon_{k_j}+\sum_{k\neq k_j} 2 v_k^2 \epsilon_k - \frac{\Delta^2}{G} -G\sum_{k\neq k_j}v_k^4+G\frac{N-S}{2}$$

$$- 4\lambda_2\sum_{k\neq k_j}(u_kv_k)^2$$ (21)

where $\lambda_2$ is the extra Lagrange multiplier introduced in the LN approach in order to take into account the particle-number fluctuation effects, S is the proton or neutron seniority for a given configuration and N is the neutron or proton number. The shell correction energy takes into account the single particle level density, which affects the binding energy. For a given nucleus, if the Fermi level is situated just above a closed shell more binding energy is observed than on average, while if it is placed just below, the binding is less than the average [24]. The shell correction energy is given by,

$$\tilde{E}_{shell-correction}=-2\int_{-\infty}^{\lambda} \mbox{\~g}(\epsilon)\epsilon d\epsilon$$ (22)

where g($\epsilon$) is the smooth part of the level density, g($\epsilon$), due to oscillations of the level density in the shell model. g($\epsilon$)d$\epsilon$ is the number of levels in the energy interval between $\epsilon$ and $\epsilon$+d$\epsilon$ and $\lambda$ is the Fermi energy.

In order to obtain the configuration dependent PES for a given multi-qp state, a process of diabatic blocking is necessary, where the given orbitals which are occupied by the specified quasiparticles are followed and blocked. In an axially-symmetric Woods-Saxon potential, the single-particle states may be specified by the $\Omega$ quantum number (the spin projection onto the symmetry axis). However, for non-axial shapes, $\Omega$ is not conserved and is no longer a good quantum number. Indeed, the only symmetries still preserved are reflection symmetry (implying conserved parity) and a rotation by 180$^o$ about the x axis (the signature quantum number $\alpha$). For even-mass nuclei $\alpha$=0 or 1, and for odd-mass nuclei $\alpha$=$\pm$1/2.

Since the quantum number $\Omega$ is no longer a good quantum number, a new way to follow a given configuration must be given. The CCBM calculates a set of average Nilsson numbers, $\langle$N$\rangle$, $\langle{n_z}\rangle$, $\langle\Lambda\rangle$ and $\langle \mid\Omega\mid \rangle$ for the configurations to be constrained in deformation space. For most of the orbitals involved in the multi-qp configurations in the mass A$\sim$130 region, these average numbers are well defined and quite stable in the deformation area considered. The nuclear shape is thus obtained by minimizing the calculated potential energy in the quadrupole ($\beta_2$, $\gamma$) deformation space, with hexadecapole ($\beta_4$) variation.