a.mean-fieldeffectiveinteractions.docVIP

The Hartree-Fock-Bogoliubov Method 1. Basics of a mean-field description The basic building block of any mean-field model is a set of single-nucleon wave functions: the number of single-particle wave functions Nwf is larger than the number of nucleons A=Z+N. Independent single-particle model: state of a nucleus is described by a Slater determinant: for occupied states for unoccupied 1≤ I ≤ A states (i A) Paring correlations concept of independent quasi-particles defined by the Bogoliubov transformation The ground state of the system is given by the condition defined as the quasi-particle vacuum: quasi-particle wave functions in coordinate space: 2. Hartree-Fock-Bogoliubov equation Ground state |Ф? of the HFB is obtained by minimization of the total energy:  with constraints on the proton and neutron numbers Minimization of the total Routhian: HFB equation Mean-field Hamiltonian and the pairing field: 1. Quasiparticle basis Фn → diagonalizes the generalized one-body matrix R 2. Canonical basis ψi → diagonalizes the one-body density p 3. Hartee-Fock basis → diagonalizes the mean-field Hamiltonian h 3. Symmetries and Constraints i) symmetries related to the shape of the nucleus – spherical, axial quadrupole, triaxial quadrupole, octupole ii) time reversal symmetry – for even-even nonrotating nuclei - creation of a quasiparticle or rotation of the nucleus breaks time-reversal symmetry. Landscape of the energy as a function of a shape degree of freedom is explored with the help of constraints. Equations of motion are obtained by minimization of a Routhian: constraint on the expectation value: Quadratic constrain