Self-consistent description of multipole strength in exotic nuclei: Method

We use the canonical Hartree-Fock-Bogoliubov basis to implement a self-consistent quasiparticle-random-phase approximation (QRPA) with arbitrary Skyrme energy density functionals and density-dependent pairing functionals. The point of the approach is to accurately describe multipole strength functions in spherical even-even nuclei, including weakly bound drip-line systems. We describe the method and carefully test its accuracy, particularly in handling spurious modes. To illustrate our approach, we calculate isoscalar and isovector monopole, dipole, and quadrupole strength functions in several Sn isotopes, both in the stable region and at the drip lines. We also investigate the consequences of neglecting the spin-orbit or Coulomb residual interactions in the QRPA.

Figure 1

Neutron quasiparticle energies $E_{\mu }^{\mathrm{check}}$ for $s_{1/2}$ states in ${}^{174}\mathrm{Sn}$, calculated by diagonalizing the HFB Hamiltonian in the canonical basis, versus the quasiparticle energies $E_{\mu }$ obtained by directly solving the HFB equations in coordinate space. Standard (solid line, diamonds) and improved (dotted line, dots) methods are used to obtain the canonical states. See text for details.

Figure 2

Isoscalar $0^{+}$ strength function in ${}^{174}\mathrm{Sn}$ for (i) the single-proton energy cutoff ${\varepsilon }_{\mathrm{crit}}=100\text{MeV}$ and the neutron-quasiparticle occupation cutoff $v_{\mathrm{crit}}^2={10}^{-8}$ (thin solid line); (ii) ${\varepsilon }_{\mathrm{crit}}=150\text{MeV}$ and $v_{\mathrm{crit}}^2={10}^{-12}$ (dotted line); and (iii) ${\varepsilon }_{\mathrm{crit}}=200\text{MeV}$ and $v_{\mathrm{crit}}^2={10}^{-16}$ (thick solid line). Results corresponding to (ii) and (iii) practically coincide.

Figure 3

Isoscalar $0^{+}$ strength function in ${}^{174}\mathrm{Sn}$ for the box radii: $R_{\mathrm{box}}=20\hspace{0.5em}\text{fm}$ (solid line) and $R_{\mathrm{box}}=25\hspace{0.5em}\text{fm}$ (dotted line). In (a), (b), and (c) the smoothing-width parameter γ is 0.5 MeV for all energies, whereas in (d), (e), and (f) $\gamma (E)$ is given by Eq. (2). We use the same three sets of cutoff conditions as in Fig. 2, namely (i) in parts (a) and (d), (ii) in parts (b) and (e), and (iii) in parts (c) and (f).

Figure 4

Isoscalar $1^{-}$ strength function in ${}^{100,120,174,176}\mathrm{Sn}$ for the corrected dipole operator in Eq. (6) (solid line) and the uncorrected operator in Eq. (5) (dotted line). The cutoff ${\varepsilon }_{\mathrm{crit}}$ is 140 MeV, and $v_{\mathrm{crit}}^2$ is $3\times {10}^{-12}$. The self-consistency of our calculations makes the solid and dotted curves coincide nearly exactly over the whole energy range.

Figure 5

Transition density of the spurious $1^{-}$ state andthe indistinguishable curve ${\mathrm{cr}}^{2}d\rho (r)/\mathrm{dr}$ (upper panel), and the difference between the two curves (lower panel). The line thickness in the upper panel is about 0.01 fm${}^{-1}.$ The constant c is fixed at $r=6.5$ fm.

Figure 6

Isoscalar $1^{-}$ strength function in ${}^{120}\mathrm{Sn}$ (left) and ${}^{174}\mathrm{Sn}$ (right) for two box radii: $R_{\mathrm{box}}=20\text{fm}$ (solid line) and $R_{\mathrm{box}}=25\text{fm}$ (dotted line). In (a) and (b) the smoothing-width parameter is constant (γ=0.5 MeV), whereas in (c) and (d) $\gamma (E)$ is given by Eq. (2).

Figure 7

Isoscalar and isovector strength functions for (a) the $0^{+}$ channel of ${}^{120}\mathrm{Sn}$, (b) the $0^{+}$ channel of ${}^{174}\mathrm{Sn}$, (c) the $2^{+}$ channel of ${}^{120}\mathrm{Sn}$, and (d) the $2^{+}$ channel of ${}^{174}\mathrm{Sn}$. The cutoff ${\varepsilon }_{\mathrm{crit}}$ is 150 MeV and $v_{\mathrm{crit}}^2$ is ${10}^{-12}$.

Figure 8

Isoscalar $0^{+}$ strength functions in ${}^{120}\mathrm{Sn}$ without the residual spin-orbit interaction in the QRPA (dashed curve), without the residual Coulomb interaction (dotted curve), and with all terms included (solid curve).

Figure 9

Isoscalar $1^{-}$ strength functions in ${}^{174}\mathrm{Sn}$ without the spin-orbit residual interaction, with and without the correction to remove spurious strength.