We present the basic concepts and recent developments in the time-dependent density functional theory (TDDFT) for describing nuclear dynamics at low energy. The symmetry breaking is inherent in nuclear energy density functionals (EDFs), which provides a practical description of important correlations at the ground state. Properties of elementary modes of excitation are strongly influenced by the symmetry breaking and can be studied with TDDFT. In particular, a number of recent developments in the linear response calculation have demonstrated their usefulness in description of collective modes of excitation in nuclei. Unrestricted real-time calculations have also become available in recent years, with new developments for quantitative description of nuclear collision phenomena. There are, however, limitations in the real-time approach; for instance, it cannot describe the many-body quantum tunneling. Thus, we treat the quantum fluctuations associated with slow collective motions assuming that time evolution of densities are determined by a few collective coordinates and momenta. The concept of collective submanifold is introduced in the phase space associated with the TDDFT and used to quantize the collective dynamics. Selected applications are presented to demonstrate the usefulness and quality of the new approaches. Finally, conceptual differences between nuclear and electronic TDDFT are discussed, with some recent applications to studies of electron dynamics in the linear response and under a strong laser field.
The neutron pairing correlation and the soft dipole excitation in medium-mass nuclei near dripline are investigated from a viewpoint of the di-neutron correlation. Numerical analyses by means of the coordinate-space HFB and the continuum QRPA methods are performed for even-even 18−24 O, 50−58 Ca and 80−86 Ni. A clear signature of the di-neutron correlation is found in the HFB ground state; two neutrons are correlated at short relative distances < ∼ 2 fm with large probability ∼ 50%. The soft dipole excitation is influenced strongly by the neutron pairing correlation, and it accompanies a large transition density for pair motion of neutrons. This behavior originates from a coherent superposition of two-quasiparticle configurations [l × (l + 1)]L=1 consisting of continuum states with high orbital angular momenta l reaching an order of l ∼ 10. It raises a picture that the soft dipole excitation under the influence of neutron pairing is characterized by motion of di-neutron in the nuclear exterior against the remaining A − 2 subsystem. Sensitivity to the density dependence of effective pair force is discussed.
We formulate a continuum linear response theory on the basis of the Hartree-FockBogoliubov formalism in the coordinate space representation in order to describe low-lying and high-lying collective excitations which couple to one-particle and two-particle continuum states. Numerical analysis is done for the neutron drip-line nucleus 24 O. A low-lying collective mode that emerges above the continuum threshold with large neutron strength is analyzed. The collective state is sensitive to the density-dependence of the pairing. The present theory satisfies accurately the energy weighted sum rule. This is guaranteed by treating the pairing selfconsistently both in the static HFB and in the dynamical linear response equation.
We analyze spatial structure of the neutron Cooper pair in superfluid low-density uniform matter by means of BCS calculations employing a bare force and the effective Gogny interaction. It is shown that the Cooper pair exhibits a strong spatial di-neutron correlation in a wide range of neutron density ρ/ρ0 ≈ 10 −4 − 0.5. This feature is related to the crossover behavior between the pairing of the weak coupling BCS type and the Bose-Einstein condensation of bound neutron pairs. We also show that the zero-range delta interaction can describe the spatial structure of the neutron Cooper pair if the density dependent interaction strength and the cut-off energy are appropriately chosen. Parameterizations of the density-dependent delta interaction satisfying this condition are discussed.
Large-amplitude collective dynamics of shape phase transition in the low-lying states of 30−36 Mg is investigated by solving the five-dimensional (5D) quadrupole collective Schrödinger equation. The collective masses and potentials of the 5D collective Hamiltonian are microscopically derived with use of the constrained Hartree-Fock-Bogoliubov plus local quasiparticle random phase approximation method. Good agreement with the recent experimental data is obtained for the excited 0 + states as well as the ground bands. For 30 Mg, the shape coexistence picture that the deformed excited 0 + state coexists with the spherical ground state approximately holds. On the other hand, large-amplitude quadrupole-shaped fluctuations dominate in both the ground and the excited 0 + states in 32 Mg, providing a picture that is different from the interpretation of the "coexisting spherical excited 0 + state" based on the naive inversion picture of the spherical and deformed configurations.Nuclei exhibit a variety of shapes in their ground and excited states. A feature of the quantum phase transition of a finite system is that the order parameters (shape deformation parameters) always fluctuate and vary with the particle number. Especially, the large-amplitude shape fluctuations play a crucial role in transitional (critical) regions. Spectroscopic studies of low-lying excited states in transitional nuclei are of great interest to observe such unique features of the finite quantum systems.Low-lying states of neutron-rich nuclei at approximately N = 20 attract great interest, as the spherical configurations associated with the magic number disappear in the ground states. In neutron-rich Mg isotopes, the increase of the excitation energy ratio E(4 1 + )/E(2 1 + ) [1-3] and the enhancement of B(E2; 2 1 + → 0 1 + ) from 30 Mg to 34 Mg [4-6] indicate a kind of quantum phase transition from spherical to deformed shapes taking place around 32 Mg. These experiments stimulate microscopic investigations on quadrupole collective dynamics unique to this region of the nuclear chart with various theoretical approaches: the shell model [7-10], the Hartree-Fock-Bogoliubov (HFB) method [11,12], the parityprojected Hartree-Fock (HF) [13], the quasiparticle random phase approximation (QRPA) [14,15], the angular-momentum projected generator coordinate method (GCM) with [16] and without [17,18] restriction to the axial symmetry, and the antisymmetrized molecular dynamics [19]. Quite recently, excited 0 + states were found in 30 Mg and 32 Mg at 1.789 MeV and 1.058 MeV, respectively, and their characters are presently under hot discussion [20-23]. For 30 Mg, the excited 0 2 + state is interpreted as a prolately deformed state which coexists with the spherical ground state. For 32 Mg, from the observed population of the excited 0 2 + state in the (t,p) reaction on 30 Mg, it is suggested [22] that the 0 2 + state is a spherical state coexisting with the deformed ground state and that their relative energies are inverted at N = 20. However, available shell-mod...
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