Implementation of nuclear time-dependent density-functional theory and its application to the nuclear isovector electric dipole resonance

Background: Time-dependent density-functional theory (TDDFT) continues to be useful in describing a multitude of low-energy static and dynamic properties. In particular, with recent advances of computing capabilities, large-scale TDDFT simulations are possible for fission dynamics as well as isovector dipole (IVD) resonances.

Purpose: Following a previous paper [Shi, Phys. Rev. C 98, 014329 (2018)], we first present a time-dependent extension of the density-functional theory to allow for dynamic calculations based on the obtained static Hartree-Fock + Bardeen-Cooper-Schrieffer (BCS) results. Second, we apply the TDDFT + BCS method to a systematic description of the IVD resonances in the Zr, Mo, and Ru isotopes.

Methods: To benchmark the TDDFT code, we compute the strengths of IVD resonances for light nuclei using two complementary methods: TDDFT and FAM-QRPA methods. For the TDDFT results, additional benchmark calculations have been performed using the well-tested code Sky3D. In these three calculations, the important ingredients which have major influence on the results, such as time-odd potentials, boundary conditions, smoothing procedures, spurious peaks, etc., have been carefully examined.

Results: The current TDDFT and the Sky3D codes yield almost identical response functions once both codes use the same time-odd mean fields and absorbing boundary conditions. The strengths of the IVD resonances calculated using the TDDFT and FAM-QRPA methods agree reasonably well with the same position of the giant dipole resonance. Upon seeing a reasonable accuracy offered by the implemented code, we perform systematic TDDFT + BCS calculations for spherical Zr and Mo isotopes near $N=50$, where experimental data exist. For neutron-rich Zr, Mo, and Ru isotopes where shape evolution exists we predict the photoabsorption cross sections based on oblate and triaxial minima.

Conclusions: The TDDFT + BCS code provides reasonable description for IVD resonances. Applying it to the spherical Zr and Mo nuclei, a reasonable agreement with experimental data has been achieved. For neutron-rich Zr isotopes, the photoabsorption cross section based on the two coexisting minima reflects the feature of the deformation of the minima. This suggests the possibility of obtaining additional information about the ground-state deformation by comparing the GDR data with the TDDFT + BCS results.

Figure 1

The response functions, $\langle \mathcal{M}(E1,\mu =0)\rangle \left(t\right)$, of ${}^{16}\mathrm{O}$ calculated with the current TDDFT code and the Sky3D code [24]. The inset of panel (b) shows the difference between the $\langle \mathcal{M}(E1,\mu =0)\rangle$ values calcualted with the TDDFT and the Sky3D codes, both including the ABC. The panel (c) shows the cross sections resulted from the respective response functions in the panel (b). The smoothing parameter is $\mathrm{\Gamma }=0$.

Figure 2

The calculated photoabsorption cross sections of ${}^{16}\mathrm{O}$ using Eq. (25). The left panel shows the results with SIII-full EDF, whereas the right panel shows those with SIII-even. The thinner black lines indicate the results without smoothing procedure.

Figure 3

The calculated photoabsorption cross sections for ${}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}$, using TDDFT of the present implementation and FAM-RPA based on the hfbtho with the SkM* and unedf1 EDFs.

Figure 4

The calculated photoabsorption cross sections for ${}^{16}\mathrm{O}$, using FAM-RPA method based on the hfbtho with the unedf1 EDF. Three different HO basis numbers are used to see the convergence of the results.

Figure 5

Time evolution of the IV density ${\rho }_p\left(\mathbf{r}\right)-{\rho }_n\left(\mathbf{r}\right)$ (in ${\mathrm{fm}}^{-3}$) in the $x-z$ plane ($y=0$) for the IVD mode in ${}^{24}\mathrm{Mg}$.

Figure 6

The calculated strength functions of ${}^{24}\mathrm{Mg}$ using SkM* and unedf1 EDFs, with the TDDFT and the FAM-RPA methods.

Figure 7

The calculated strength functions of ${}^{34}\mathrm{Mg}$ with SkM* EDF using the TDDFT + BCS and the FAM-QRPA methods.

Figure 8

The IV $E1$ cross sections calculated using the SkM* and SLy4 EDFs with the smoothing parameters $\mathrm{\Gamma }=1.0$ and 2.0 MeV. The experimental data are from Ref. [68], which are extracted from Refs. [69, 70].

Figure 9

Similar to Fig. 8, except that the calculations are performed with unedf1 EDF for $\mathrm{\Gamma }=0.5$, 1.0, and 2.0 MeV. A photoabsorption cross section calculated without pairing with $\mathrm{\Gamma }=0.5$ MeV is also shown.

Figure 10

Calculated photoabsorption cross sections using unedf1 EDF, with smoothing parameter $\mathrm{\Gamma }=2.0$ MeV. The experimental data for GDR (red and black crosses) are from Refs. [68, 72]. In the third row, the same cross sections for Mo isotopes are plotted in the logarithmic scale. The data for the lower-energy parts (red crosses) are from Refs. [73, 74, 75, 76, 77]. The numbers are extracted from Refs. [69, 70].

Figure 11

Calculated potential-energy surfaces for ${}^{98-108}\mathrm{Zr}$, using the unedf1 EDF. The energies are in MeV.

Figure 12

The same as described in the caption of Fig. 11, except for ${}^{100-110}\mathrm{Mo}$.

Figure 13

The same as described in the caption of Fig. 11, except for ${}^{102-112}\mathrm{Ru}$.

Figure 14

The cross sections of ${}^{100}\mathrm{Mo}$ calculated with unedf1 EDF, $\mathrm{\Gamma }=2.0$ MeV. The black solid line corresponds to the spherical ground state as shown in Fig. 12. The green dashed line corresponds to the triaxial minimum in the energy surface of ${}^{100}\mathrm{Mo}$, with ($Q_{20}$, $Q_{22})=(3.5,2.4)\mathrm{b}$. The above two curves are based on results without constraints on the quadrupole moments. The red dotted line corresponds to a prolate deformation which has been constrained to have $Q_{20}=5.0$ b. The experimental data are extracted from Refs. [69, 70].

Figure 15

Photoabsorption cross sections calculated for ${}^{100-108}\mathrm{Zr}$ with unedf1 EDF with smoothing parameter $\mathrm{\Gamma }=2.0$ MeV (thick line) and 1.0 MeV (thin line). The plots in upper row correspond to the results based on the prolate minima for ${}^{100,102}\mathrm{Zr}$, and the triaxial minima for ${}^{104-108}\mathrm{Zr}$; the plots in the lower row correspond to those based on the oblate minima, see Fig. 11. The thinner green lines indicate results with $\mathrm{\Gamma }=1.0$ MeV.

Figure 16

The fraction of the strengths for IVD resonances below $E_c=10$ MeV [Eq. (32)] for zirconium isotopes with spherical, prolate(for ${}^{100,102}\mathrm{Zr}$)/triaxial(for ${}^{104,106,108}\mathrm{Zr}$), and oblate deformations. The smoothing parameter $\mathrm{\Gamma }=0.5$ MeV is used in the calculation. The integrations for the total strengths are taken from 0 to 80 MeV.

Figure 17

The same as described in the caption of Fig. 15, except for ${}^{102-110}\mathrm{Mo}$ and ${}^{104-112}\mathrm{Ru}$.

Figure 18

The upper panel are the energy differences of peaks corresponding to the vibrations along the $y-$ and $z-\mathrm{axis}$ ($E_y-E_z$), as well as along the $x-$ and $y-\mathrm{axis}$ for $\gamma =0^{\circ }-{60}^{\circ }$ of ${}^{106}\mathrm{Mo}$. The lower panel are the calculated relative cross sections corresponding to the energies in the upper panel, that is, ${\sigma }_{\mathrm{x},\mathrm{y},\mathrm{z}}\equiv {\sigma }_{\mathrm{abs}.}\left(E_{\mathrm{x},\mathrm{y},\mathrm{z}}\right)$.