Time-dependent density-functional description of nuclear dynamics

The basic concepts and recent developments in the time-dependent density-functional theory (TDDFT) for describing nuclear dynamics at low energy are presented. The symmetry breaking is inherent in nuclear energy density functionals, 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 the 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, the quantum fluctuations associated with slow collective motions are explicitly treated assuming that time evolution of densities is 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.

Figure 1

Potential energy curves as functions of deformation parameter $\beta$. a corresponds to spherical nuclei, b and c correspond to transitional nuclei, and d to well-deformed nuclei. From [4].

Figure 2

Deformed nuclei in different time scales: (a) shorter than the symmetry-restoring time ${\tau }_{\mathrm{SSB}}$, and (b) larger than ${\tau }_{\mathrm{SSB}}$.

Figure 3

Second-order diagrams for particle-vibration coupling, contributing to (a) the self-energy part ${\mathrm{\Sigma }}_{ki}(E)$ and (b) effective two-particle interactions.

Figure 4

Deformation and photoabsorption in Nd and Sm isotopes, calculated with the Skyrme energy functional of ${\mathrm{SkM}}^{*}$. Calculated and experimental intrinsic quadrupole moments are denoted by open and closed symbols, respectively, for (a) Nd and (b) Sm isotopes. For spherical nuclei with $N=82$ and 84, we also plot the values (triangles) extracted from the QRPA calculation for $B(E2;2^{+}\to 0^{+})$. Photoabsorption cross sections for (c) Nd and (d) Sm isotopes. The solid lines show the calculation and the solid (blue) symbols are experimental data ([59], [60]). Adapted from [413].

Figure 5

The excitation energies of the isoscalar giant monopole resonances in the Sm isotopes; calculated values (solid symbols) and experimental data (open symbols) ([159]). From [414].

Figure 6

Neutron capture cross sections $n+{}^{141}\mathrm{Sn}$ as functions of neutron energy $e$, for incident neutrons in $s_{1/2}$, $d_{3/2}$, and $d_{5/2}$ states. The Skyrme SLy4 EDF and the density-dependent pairing EDF are used. The dotted lines indicate the power-law scaling $\propto e^{l-1/2}$. From [213].

Figure 7

Internucleus potential for ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ for different energies. The horizontal axis is the distance between two ${}^{40}\mathrm{Ca}$ nuclei. The green line indicates the Coulomb potential assuming the point charge $20e$ at the center of each nucleus. DC-TDHF is an abbreviation for the density-constrained TDHF. From [379].

Figure 8

Excitation energy $E^{*}$ (solid lines) in Eq. (72) for ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ collision at different energies. The dashed lines indicate the collective kinetic energy $E_{\text{kin}}$ in Eq. (74). From [379].

Figure 9

Internucleus potential for ${}^{96}\mathrm{Zr}+{}^{124}\mathrm{Sn}$ calculated at $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=230\mathrm{MeV}$. The solid red curve is based on Eq. (75), while the dashed blue line is the one of the density-constrained calculation of Eq. (73). The frozen density potential is plotted by the solid circles. DD-TDHF is an abbreviation for the dissipative-dynamics TDHF. From [392].

Figure 10

Transfer cross sections for ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\text{lab}}=174\mathrm{MeV}$. The solid red lines show the TDDFT (TDHF) results with the particle-number projection. The experimental data and the grazing results are taken from [71]. See text for details. From [319].

Figure 11

The fluctuation in the proton number distribution ${\sigma }_{ZZ}\equiv (⟨Z^2⟩-⟨Z{⟩}^2{)}^{1/2}$ for final fragments in the ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ reaction. The horizontal axis corresponds to the scattering angle that is determined by the impact parameter in the TDDFT (TDHF) calculation. The solid red line shows results based on Eq. (82) to produce larger fluctuations than the dashed blue line (TDDFT). The circles show experimental data ([300]). From [334].

Figure 12

Schematic illustrations of collective submanifold in the TDHFB space. (a) 1D collective path specified by a series of the states $|\varphi (q)⟩$ with a local generator $P(q)=id/dq$. (b) A section of a 2D collective hypersurface and local generators of the collective coordinate $\mathbf{(}{\widehat{P}}_1(q),{\widehat{P}}_2(q)\mathbf{)}$.

Figure 13

Low-lying spectra and $B(E2)$ values in $e^2{\mathrm{fm}}^4$ for ${}^{76}\mathrm{Kr}$; (a) experimental data from [69], (b) the relativistic GCM calculations in the $(\beta ,\gamma )$ plane with the particle-number and the angular-momentum projections, and (c) the calculation of the 5D collective Hamiltonian using the cranking inertial masses (Sec. 5d1). From [408].

Figure 14

Application of the ASCC method to the oblate-prolate shape coexistence phenomenon in ${}^{68}\mathrm{Se}$. (a) The collective path for ${}^{68}\mathrm{Se}$ obtained by the ASCC method. The solid (red) line shows the collective path running along the valley of the potential energy surface projected on the $(\beta ,\gamma )$ deformation plane. (b) Vibrational wave functions squared of the lowest (left) and the second-lowest states (right) for each angular momentum. In each panel, different $K$ components of the vibrational wave functions and the sum of them are plotted as functions of $\gamma (q)$. For excitation spectra, see Fig. 15. Adapted from [140].

Figure 15

Application of the LQRPA method to the oblate-prolate shape coexistence and fluctuation phenomenon in ${}^{68}\mathrm{Se}$. (a) Collective inertial masses $D_{\beta \beta }(\beta ,\gamma )$ and $D_{\gamma \gamma }(\beta ,\gamma )$, (b) excitation spectrum, and (c) vibrational wave functions ${\beta }^4\sum _K|{\mathrm{\Phi }}_{IKk}(\beta ,\gamma ){|}^2$. Adapted from [141].