Thouless-Valatin rotational moment of inertia from linear response theory

Spontaneous breaking of continuous symmetries of a nuclear many-body system results in the appearance of zero-energy restoration modes. These so-called spurious Nambu-Goldstone modes represent a special case of collective motion and are sources of important information about the Thouless-Valatin inertia. The main purpose of this work is to study the Thouless-Valatin rotational moment of inertia as extracted from the Nambu-Goldstone restoration mode that results from the zero-frequency response to the total-angular-momentum operator. We examine the role and effects of the pairing correlations on the rotational characteristics of heavy deformed nuclei in order to extend our understanding of superfluidity in general. We use the finite-amplitude method of the quasiparticle random-phase approximation on top of the Skyrme energy density functional framework with the Hartree-Fock-Bogoliubov theory. We have successfully extended this formalism and established a practical method for extracting the Thouless-Valatin rotational moment of inertia from the strength function calculated in the symmetry-restoration regime. Our results reveal the relation between the pairing correlations and the moment of inertia of axially deformed nuclei of rare-earth and actinide regions of the nuclear chart. We have also demonstrated the feasibility of the method for obtaining the moment of inertia for collective Hamiltonian models. We conclude that from the numerical and theoretical perspective, the finite-amplitude method can be widely used to effectively study rotational properties of deformed nuclei within modern density functional approaches.

Figure 1

Thouless-Valatin rotational moment of inertia in (a) erbium and (b) ytterbium isotopic chains. Full lines indicate the volume pairing and vanishing pairing options; in the latter case, the values of the TV inertia are divided by two in order to make a comparison with other results. The dashed lines show the experimental data from the rotational bands $(2^{+}$ states). In addition, panels show Belyaev and rigid body (divided by 2) values for the moment of inertia. For erbium isotopes, the volume pairing was adjusted to the experimental proton and neutron pairing gaps of ${}^{166}\mathrm{Er}$, while for ytterbium isotopes, the pairing gaps of ${}^{168}\mathrm{Yb}$ were chosen.

Figure 2

The same as Fig. 1 but for (a) uranium and (b) plutonium isotopic chains. For uranium isotopes, the volume pairing was adjusted to the experimental proton and neutron pairing gaps of ${}^{238}\mathrm{U}$, while for plutonium isotopes, the pairing gaps of ${}^{240}\mathrm{Pu}$ were chosen.

Figure 3

The flow (lines with arrows) of the induced isoscalar current density ${\vec{j}}_0$ and its magnitude (color contours and thickness of the flow line) for ${}^{166}\mathrm{Er}$ with volume pairing setup, obtained from the response to the ${\widehat{J}}_y$ operator, are shown in the $x\text{−}z$ plane. The full cross section (a) shows the total current density, while the half plots (b)–(e) indicate the partial induced currents as given by the specific QRPA blocks. The black dashed contour line indicates the nuclear surface at matter density of ${\rho }_0=0.08{\mathrm{fm}}^{-3}$. See the text for further details.

Figure 4

Same as Fig. 3, but with vanishing pairing.

Figure 5

Calculated Thouless-Valatin rotational moment of inertia and the HFB binding energy of ${}^{166}\mathrm{Er}$, both as function of the deformation parameter.