Global performance of multireference density functional theory for low-lying states in $sd$-shell nuclei

We present a comprehensive study of low-lying states in even-even Ne, Mg, Si, S, Ar isotopes with the multireference density functional theory (MR-DFT) based on a relativistic point-coupling energy density functional (EDF). Beyond mean-field (BMF) effects are taken into account by configuration mixing of both particle-number and angular-momentum projected axially deformed states with generator coordinate method. Global performance of the MR-DFT for the properties of both ground state and of the first $2^{+},4^{+}$ states is examined, in comparison with previous studies based on nonrelativistic EDFs and available data. Our results indicate that an EDF parametrized at the BMF level is demanded to achieve a quantitative description.

Figure 1

The energy of mean-field states in ${}^{44}\mathrm{S}$ and that of projected states with projection onto particle number and angular momentum $\left(J=0\right)$ as functions of axial deformation parameter $\beta$. The calculations are performed by expanding the Dirac spinors of nucleons in the harmonic oscillator basis with different number of major shells.

Figure 2

Low-lying spectra of ${}^{44}\mathrm{S}$ calculated with the MR-CDFT using the PC-PK1 and PC-F1 forces, in comparison with available data [16]. The numbers on the arrow are the $B\left(E2\right)$ values (in units of $e^2{\mathrm{fm}}^4)$.

Figure 3

Dynamical correlation energies of $sd$-shell nuclei from the configuration-mixing GCM calculation using the PC-F1 force, in comparison with the similar calculation using the Skyrme SLy4 force [78]. The results by the cranking prescription based on the mean-field ground-state solution by the PC-F1 force are also given for comparison.

Figure 4

(a) Energy per nucleon for Ne, Mg, Si, S, and Ar isotopes in the $sd$-shell region from both mean-field and configuration-mixing GCM calculations using the PC-F1 force, in comparison with data. (b) Energy per nucleon for silicon isotopes calculated using the relativistic PC-F1 and PC-PK1 forces in comparison with those by the nonrelativistic SLy4 force from Ref. [78] and available data from Ref. [80]. The MF results of nonrelativistic SLy4 include the correlation energy gained from the particle number projection.

Figure 5

(a) Two-neutron separation energy as a function of neutron number. The inset panel shows that of $N=20$, 22, and 28 isotonic chains. (b) Neutron $N=20$ shell gap by the mean-field (MF), PNP $\left(N\&Z\right)$, PN1DAMP $\left(J=0\right)$, and full GCM calculations using the PC-F1 force as a function of proton number. Data are taken from Ref. [80].

Figure 6

Charge density distributions calculated with the spherical configuration, the configuration corresponding to the global minimum of $J=0$ energy curve (Min.), and the full configurations (GCM), respectively. The charge density units are in ${\mathrm{fm}}^{-3}$. The experimental data from Ref. [83] are given for comparison.

Figure 7

Root-mean-square charge radii of $sd$-shell even-even nuclei as functions of neutron number. The results for Ne, Mg, Si, S, and Ar isotopes obtained by PN1DAMP+GCM (red line) are compared with those from the RMF+BCS mean-field (blue line) using the same relativistic energy density functional. The available data for Mg isotopes from Ref. [84] and others from Ref. [85] are also given for comparison.

Figure 8

Root-mean-square charge radii of Si isotopes from both mean-field and configuration-mixing GCM calculations using the relativistic PC-F1 and PC-PK1 forces as functions of neutron number, in comparison with those by the SLy4 force calculations on good particle number $\left(N\&Z\right)$ from Ref. [78] and available data from Ref. [85].

Figure 9

Changes $\delta r_c^2$ in the mean-square charge radii of $sd$-shell even-even nuclei due to the dynamical correlation effects, where $\delta r_c^2=r_c^2\left(0_1^{+}\right)-r_c^2\left({\beta }_m\right)$ with $r_c^2\left(0_1^{+}\right)$ and $r_c^2\left({\beta }_m\right)$ being the mean-square radius of the $0_1^{+}$ state and the mean-field ground state, respectively. The $r_c^2\left({\beta }_m\right)$ by the SLy4 force is the mean-square radius of the minimum of the particle number projected energy curve, while that by the PC-F1 force is the minimum of the mean-field energy curve. All the results obtained by the SLy4 are taken from Ref. [78].

Figure 10

Central depletion factor of proton (a)–(e) and charge (f)–(j) in $sd$-shell even-even isotopes as functions of neutron number. The results are obtained from spherical state (blue line), minimum of projected $(N\&Z,J=0)$ energy curve (olive line), and the full configuration mixing (red line) calculation using the PC-F1, respectively.

Figure 11

Excitation energies of the $2_1^{+}$ states (a) and $B(E2;0_1^{+}\to 2_1^{+})$ values (b) in Mg isotopes as functions of neutron number. The results from PN1DAMP+GCM (solid line) are compared with 1DAMP+GCM (dash line) [53] calculation using the same relativistic density functional PC-F1. Available data [90] are also given for comparison.

Figure 12

Excitation energies of the $2_1^{+}$ state (left panel) and $4_1^{+}$ state (right panel) as a function of neutron number in $sd$-shell even-even Ne, Mg, Si, S, and Ar isotopes, in comparison with available data from Refs. [90, 91, 92, 93].

Figure 13

Excitation energy ratios $R_{42}=E\left(4_1^{+}\right)/E\left(2_1^{+}\right)$ as a function of neutron number in $sd$-shell even-even Ne, Mg, Si, S, and Ar isotopes, in comparison with available data from Refs. [90, 91, 92, 93].

Figure 14

Electric quadrupole transition strengths $B(E2;0_1^{+}\to 2_1^{+})$ as functions of the neutron number in $sd$-shell even-even isotopes. The results obtained from BMF (solid line) are compared with those values derived from the deformation ${\beta }_m$ predicted by MF (dash line) using the same relativistic density functional PC-F1. Data are taken from Refs. [90, 95]. More details see the text.

Figure 15

Spectroscopic quadrupole moments $Q^{\mathtt{spec}}$ of the first $2^{+}$ as a function of neutron number in $sd$-shell even-even isotopes, in comparison with available data from Ref. [96].

Figure 16

(a) Excitation energy $E_x\left(2_1^{+}\right)$ and (b) $B(E2;0_1^{+}\to 2_1^{+})$ value as functions of proton number in $N=28$ isotones. The results of the Gogny D1S from Ref. [30] were obtained without PNP. The inset panel shows the ratio $R_{42}$. Data are taken from Ref. [90].

Figure 17

Neutron-proton decoupling factor $\eta$ [cf. Eq. (23)] of the $2_1^{+}$ state of even-even $sd$-shell nuclei as a function of neutron number, in comparison with available data from Refs. [98, 99, 100, 101, 102, 103, 104]. The experimental data for ${}^{28,30}\mathrm{Ne}$ and ${}^{32,34}\mathrm{Mg}$ are extracted using the formula and corresponding parameters given in Ref. [105].

Figure 18

Longitudinal C2 form factor $|F_2{\left(q\right)|}^2$ for the transition from the ground state $0_1^{+}$ to the excited state $2_1^{+}$ for ${}^{24}\mathrm{Mg}$. The inset panel shows the corresponding transition densities. Data are taken from Ref. [106] (squares and circles).

Figure 19

Same as Fig. 18, but for ${}^{28}\mathrm{Si}$. Data are taken from Refs. [106, 107].

Figure 20

Same as Fig. 18, but for ${}^{32}\mathrm{S}$.

Figure 21

(a) Calculated $E\left(2_1^{+}\right)$ excitation energies of 48 even-even nuclei in the $sd$-shell region as a function of their experimental values. (b) Calculated $B(E2;0_1^{+}\to 2_1^{+})$ transition strengths of 38 even-even nuclei in the $sd$-shell region as a function of their experimental values. Experimental data are taken from Ref. [90].

Figure 22

Histogram of the logarithmic errors $R$ defined in Eq. (27). (a): Excitation energies of the first $2^{+}$ states for 48 even-even nuclei. (b): $B(E2;0_1^{+}\to 2_1^{+})$ transition strengths of 38 even-even nuclei in the $sd$-shell region. The distribution of $B\left(E2\right)$ without experimental error bar is also given (dash line). Experimental data are taken from Ref. [90].