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Physical Review C

Damping of collective states in an extended random-phase approximation with ground-state correlations

M. Tohyama

Published Apr 24, 2007• 24 citations
Publisher

Abstract

Applications of an extended version of the Hartree-Fock theory and the random-phase approximation derived from the time-dependent density-matrix theory (TDDM) are presented. In this TDDM-based theory, the ground state is given as a stationary solution of the TDDM equations and the excited states are calculated using the small-amplitude limit of TDDM. The first application presented is an extended Lipkin model in which an interaction term describing a particle scattering is added to the original Hamiltonian so that the damping of a collective state is taken into account. It is found that the TDDM-based theory well reproduces the ground state and excited states of the extended Lipkin model. The quadrupole excitation of the oxygen isotopes $^{16,20,22}\mathrm{O}$ is also studied as realistic applications of the TDDM-based theory. It is found that large fragmentation of the giant quadrupole resonance in $^{16}\mathrm{O}$ is reproduced, and it is pointed out that the effects of ground-state correlations are quite important for fragmentation. It is also found that the quadrupole states in neutron-rich oxygen isotopes have small spreading widths.

DOI
10.1103/physrevc.75.044310

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Applications of an extended version of the Hartree-Fock theory and the random-phase approximation derived from the time-dependent density-matrix theory (TDDM) are presented. In this TDDM-based theory, the ground state is given as a stationary solution of the TDDM equations and the excited states are calculated using the small-amplitude limit of TDDM. The first application presented is an extended Lipkin model in which an interaction term describing a particle scattering is added to the original Hamiltonian so that the damping of a collective state is taken into account. It is found that the TDDM-based theory well reproduces the ground state and excited states of the extended Lipkin model. The quadrupole excitation of the oxygen isotopes 16,20,22O^{16,20,22}\mathrm{O}16,20,22O is also studied as realistic applications of the TDDM-based theory. It is found that large fragmentation of the giant quadrupole resonance in 16O^{16}\mathrm{O}16O is reproduced, and it is pointed out that the effects of ground-state correlations are quite important for fragmentation. It is also found that the quadrupole states in neutron-rich oxygen isotopes have small spreading widths.
The ground states and excited states of the Lipkin model Hamiltonian are calculated using a new theoretical approach that has been derived from an extended time-dependent Hartree-Fock theory known as the time-dependent density-matrix (TDDM) theory. TDDM theory enables us to calculate correlated ground states, and its small amplitude limit (STDDM), which is a version of extended RPA theories based on a correlated ground state, can be used to calculate excited states. It is found that this TDDM plus STDDM approach gives much better results for both the ground states and the excited states than the Hartree-Fock ground state plus RPA approach.
A correlated ground state of 16O{}^{16}\mathrm{O}16O and an E2E2E2 giant resonance built on it are calculated using an extended version of the time-dependent Hartree-Fock theory called the time-dependent density-matrix (TDDM) theory. The Skyrme III force is used as an effective interaction for the calculation of both a mean field and two-body correlations. It is found that a TDDM calculation with the Skyrme force gives reasonable ground-state correlations and a large spreading width of the E2E2E2 giant resonance.
The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of 40^{40}40Ca and 48^{48}48Ca and the other the fusion reactions of 16^{16}16O + 16^{16}16O.
The time-dependent density-matrix theory (TDDM) gives a correlated ground state as a stationary solution of the time-dependent equations for one-body and two-body density matrices. The small amplitude limit of TDDM (STDDM) is a version of extended RPA theories which include the effects of ground state correlations. It is shown that the solutions of the Hartree-Fock Bogoliubov theory and the quasi-particle RPA satisfy the TDDM and STDDM equations, respectively, when only pairing-type correlations are taken into account in TDDM and STDDM.
We propose a time-independent method for finding a correlated ground state of an extended time-dependent Hartree-Fock theory, known as the time-dependent density-matrix theory (TDDM). The correlated ground state is used to formulate the small amplitude limit of TDDM (STDDM) which is a version of extended RPA theories with ground-state correlations. To demonstrate the feasibility of the method, we calculate the ground state of 22O and study the first 2+ state and its two-phonon states using STDDM.
Taking 16O as an example of realistic nuclei, we demonstrate that a correlated ground state can be obtained as a long time solution of a time-dependent density-matrix formalism (TDDM) when the residual interaction is adiabatically treated. We also study in TDDM the E2 giant resonance of 16O built on the correlated ground state and compare it with that built on the Hartree-Fock ground state. It is found that a spurious mixing of low frequency components seen in the latter is eliminated by using the correlated ground state.