Triaxial Relativistic Hartree-Bogoliubov Results with the PC-PK1 Density Functional
Theoretical Framework
The relativistic Hartree-Bogoliubov (RHB) theory provides a unified and self-consistent treatment of mean fields and pairing correlations,
and it is formalized by the RHB euqation
where ${h_D}$ is the single-nucleon Dirac Hamiltonian, ${\Delta}$ is the pairing field, $U_k$ and $V_k$ are the quasiparticle wavefunctions, and $E_k$ denotes the quasiparticle energies.
The Dirac Hamiltonian reads
where the scalar field $S$ and the vector field $V$ are coupled with the corresponding scalar and vector densities in a self-consistent way [1,2,3,4].
The matrix elements of pairing field ${\Delta}$ can be written as
where $\kappa=V^*U^T$ is the pairing tensor and $V^{pp}$ is the pairing force.
For present results, the PC-PK1 relativistic density functional [5] and a finite-range separable pairing force with the pairing strength $G=728 \mathrm{MeV\cdot fm}^3$ [6] are employed.
The potential energy surface (PES) with respect to the quadrupole deformation parameters $\beta$ and $\gamma$ can be obtained by constraint calculations against mass quadrupole moments [5,6],
and the lowest energy state in the PES provides the ground state at the mean field level.
The five-dimensional collective Hamiltonian is employed to extend the mean-field solutions to describe spectroscopic properties
and to bring beyond-mean-field dynamical correlation energies to nuclear ground states.
The 5DCH is expressed in terms of the two quadrupole deformation parameters $\beta$ and $\gamma$ and three Euler angles $(\phi,\theta,\psi)\equiv\Omega$ that define the orientation of the intrinsic principal axes in the laboratory frame,
It includes the vibrational kinetic energy ${T}_{\mathrm{vib}}(\beta,\gamma)$, rotational kinetic energy ${T}_{\mathrm{rot}}$, and the collective potential $V_{\mathrm{coll}}(\beta,\gamma)$, and their expressions can be seen in Refs. [7,8]. The collective parameters including the moments of inertia $\mathcal{I}_k$, the mass parameters $B_{\beta\beta}$, $B_{\beta\gamma}$, $B_{\gamma\gamma}$, and the collective potential $V_{\mathrm{coll}}(\beta,\gamma)$ governs the entire dynamics of the 5DCH. As in Refs.[5,6], the collective parameters are determined by the self-consistent RHB calculations with no adjustable parameters.
The moments of inertia are computed by the Inglis-Belyaev formula, the mass parameters are calculated in the cranking approximation, and the collective potential is obtained by subtracting the energy of the zero-point motion from the total energy surface.
Diagonalization of the 5DCH yields the energy of the collective ground state $0_1^+$ and other low-lying excited states.
