Pairing and rotational properties of actinides and superheavy nuclei in covariant density functional theory

The cranked relativistic Hartree-Bogoliubov theory has been applied for a systematic study of pairing and rotational properties of actinides and light superheavy nuclei. Pairing correlations are taken into account by the Brink-Booker part of finite-range Gogny D1S force. For the first time, in the covariant density functional theory (CDFT) framework, the pairing properties of deformed nuclei are studied via the quantities (such as three-point ${\Delta }^{\left(3\right)}$ indicators) related to odd-even mass staggerings. The investigation of the moments of inertia at low spin and the ${\Delta }^{\left(3\right)}$ indicators shows the need for an attenuation of the strength of the Brink-Booker part of the Gogny D1S force in pairing channel. The investigation of rotational properties of even-even and odd-mass nuclei at normal deformation, performed in the density functional theory framework in such a systematic way for the first time, reveals that in the majority of the cases the experimental data are well described. These include the evolution of the moments of inertia with spin, band crossings in the $A\ge 242$ nuclei, the impact of the particle in specific orbital on the moments of inertia in odd-mass nuclei. The analysis of the discrepancies between theory and experiment in the band crossing region of $A\le 240$ nuclei suggests the stabilization of octupole deformation at high spin, not included in the present calculations. The evolution of pairing with deformation, which is important for the fission barriers, has been investigated via the analysis of the moments of inertia in the superdeformed minimum. The dependence of the results on the CDFT parametrization has been studied by comparing the results of the calculations obtained with the NL1 and NL3* parametrizations.

Figure 1

Scaling factors as a function of neutron number for individual nuclei of different isotope chains. The dotted line corresponds to the average scaling factor $f_{\mathrm{av}}$ for a given parametrization of the CDFT. The results of the calculations with the NL1 and NL3* parametrizations are presented.

Figure 2

Calculated and experimental moments of inertia at low spin. Experimental moments of inertia are extracted from the energies of the $2^{+}$ states. Calculated values are obtained in the CRHB+LN calculations with $f_{\mathrm{av}}$ specific to a given parametrization at the rotational frequency corresponding to experimental energy of the $2^{+}\to 0^{+}$ transition. Experimental data are shown by filled black circles, while calculated values are shown with red triangles (the NL1 parametrization) and green squares (the NL3* parametrization). Theoretical results are shown at ${\Omega }_x=0.02$ MeV in the cases when experimental data are not available.

Figure 3

The calculated (lines) and experimental (circles) quadrupole deformation parameters ${\beta }_2$. The experimental values of ${\beta }_2$ obtained in the direct measurements [70] are shown by solid circles, while those deduced from the $2^{+}\to 0^{+}$ transition energies, with the prescription of Ref. [72], are given by open circles. The results of the calculations with the NL1 and NL3* parametrizations are shown by red dashed and green solid lines, respectively.

Figure 4

The same as in Fig. 2 but for the CRHB calculations with scaling factor $f=1.0$ and only for the nuclei in which experimental data are available. To make the comparison with Fig. 2 easier, the difference between the highest and lowest values on the vertical axis is kept the same as in Fig. 2; the only exception is the panel with the Cm isotopes. As a consequence, the different ranges for vertical axis are used for top and bottom panels. Moreover, the different ranges are used for vertical axis in the case of the Cm and Pu isotopes.

Figure 5

Experimental and calculated neutron three-point indicators ${\Delta }_{\nu }^{\left(3\right)}\left(N\right)$ as a function of neutron number $N$. The results of the CRHB and CRHB+LN calculations with the NL1 parametrization are shown.

Figure 6

The same as in Fig. 5 but for proton three-point indicators ${\Delta }_{\pi }^{\left(3\right)}\left(Z\right)$ as a function of proton number $Z$.

Figure 7

The same as in Fig. 5 but for the NL3* parametrization. Theoretical values have been obtained within the CRHB framework.

Figure 8

The same as in Fig. 6 but for the NL3* parametrization. Theoretical values have been obtained within the CRHB framework.

Figure 9

The experimental and calculated moments of inertia $J^{\left(1\right)}$ as a function of rotational frequency ${\Omega }_x$. The calculations are performed with the NL3* parametrization of CDFT. Calculated results and experimental data are shown by black lines and red small solid circles, respectively. Although some calculations suggest that ${}^{228}$Th is octupole soft (see Fig. 7 in Ref. [42]), its moment of inertia is rather well described in our calculations with no octupole deformation.

Figure 10

The same as in Fig. 9 but for the calculations with the NL1 parametrization.

Figure 11

Calculated proton and neutron contributions to kinematic moments of inertia $J^{\left(1\right)}$ as a function of rotational frequency ${\Omega }_x$. The calculations are performed with the NL3* parametrization of CDFT.

Figure 12

The experimental and calculated kinematic moments of inertia $J^{\left(1\right)}$ of ground-state rotational bands in ${}^{242,244}$Pu and ${}^{248}$Cm as a function of rotational frequency ${\Omega }_x$. The calculations are performed with the NL1 parametrization.

Figure 13

The same as in Fig. 12 but for the results obtained with the NL3* parametrization.

Figure 14

The chart of nuclei investigated in the current work. Solid blue and red shaded circles indicate studied odd-neutron and odd-proton nuclei, respectively. The nucleus has $Z+1$ protons and $N$ neutrons if its circle is located between the $(Z,N)$ and $(Z+2,N)$ boxes. Alternatively, it has $Z$ protons and $N+1$ neutrons if its circle is located between the $(Z,N)$ and $(Z,N+2)$ boxes.

Figure 15

Proton and neutron single-particle energies in ${}^{244}$Cm as a function of quadrupole deformation ${\beta }_2$ obtained in the calculations with the NL1 and NL3* parametrizations. Solid black and red dashed lines are used for positive- and negative-parity states, respectively. The energy of the Fermi level is shown by the blue dotted line. Deformed single-particle orbitals of interest are labeled by the Nilsson quantum numbers $\Omega \left[N{n}_z\Lambda \right]$.

Figure 16

Calculated and experimental kinematic moments of inertia $J^{\left(1\right)}$ of the indicated one-quasiproton configurations in the ${}^{241}$Am nucleus and ground-state rotational band in the reference even-even ${}^{240}$Pu nucleus. Experimental data are shown in the middle panel, while the results of the CRHB+LN calculations with the NL1 and NL3* parametrizations are shown in the left and right panels, respectively. The same symbols/lines are used for the same theoretical and experimental configurations. The symbols are used only for the configurations in odd-mass nucleus; the ground-state rotational band in the reference even-even nucleus is shown by the solid black line. The label with the structure “Odd nucleus $=$ reference even$+$even nucleus $+$ proton($\pi$)/neutron($\nu$)” is used in order to indicate the reference even-even nucleus and the type of the particle (proton or neutron) active in odd-mass nucleus. The experimental data are from Refs. [94, 99].

Figure 17

Calculated proton and neutron pairing energies in ground-state rotational band of ${}^{240}$Pu and one-quasiparticle rotational bands of ${}^{241}$Am. Thick and thin lines are used for the $(r=-i)$ and $(r=+i)$ branches of one-quasiparticle configurations, respectively. Note that neutron pairing almost does not depend on the signature of blocked proton orbital. As a result, only the $(r=-i)$ branches are shown in panel (a).

Figure 18

The same as in Fig. 16 but for ${}^{237}$Np. The experimental data are from Ref. [99].

Figure 19

The same as in Fig. 16 but for ${}^{251}$Md. Experimental data are taken from Refs. [65, 104].

Figure 20

The same as in Fig. 16 but for ${}^{235}$Np. Experimental data are taken from Ref. [105]. See text for details.

Figure 21

The same as in Fig. 16 but for ${}^{247}$Cm. Experimental data are taken from Ref. [87].

Figure 22

The same as in Fig. 16 but for ${}^{249}$Cf. Experimental data are taken from Ref. [87].

Figure 23

The same as in Fig. 16 but for ${}^{253}$No. Experimental data are taken from Refs. [66, 72].

Figure 24

The same as in Fig. 16 but for ${}^{237}$U.

Figure 25

The same as in Fig. 16 but for ${}^{239}$Pu.

Figure 26

The same as in Fig. 16 but for ${}^{249}$Cm. Experimental data are taken from Ref. [87].

Figure 27

The same as in Fig. 16 but for ${}^{243}$Pu.

Figure 28

Experimental and calculated kinematic $\left(J^{\left(1\right)}\right)$ and dynamic $\left(J^{\left(2\right)}\right)$ moments of inertia of SD rotational bands in ${}^{236,238}$U and ${}^{240}$Pu. The notation of the lines and symbols is given in the figure.

Figure 29

The single-particle levels in the superdeformed minimum of ${}^{236}$U obtained in the calculations with the NL1 and NL3* parametrizations. They are labeled by the Nilsson labels $\Omega \left[N{n}_z\Lambda \right]$ when the squared amplitude of the dominant component of the wave function exceeds 0.5. Otherwise, the states are labeled by $\Omega \left[N\right]$, where $N$ represents the dominant principal quantum number.

Figure 30

Charge quadrupole moments $Q$ [panel (a)] and kinematic moments of inertia $J^{\left(1\right)}$ [panel (b)] of the yrast SD bands as a function of neutron number $N$. The results are obtained in the CRHB+LN(NL3*) calculations at ${\Omega }_x=0.01$ MeV.

Figure 31

Kinematic moments of inertia [panel (a)] of ground-state bands of superheavy nuclei calculated at ${\Omega }_x=0.02$ MeV and their quadrupole deformations ${\beta }_2$ [panel (b)] as a function of neutron number $N$.