Elimination of spurious modes within quasiparticle random-phase approximation

We suggest a generalized method for elimination of spurious admixtures (SA) from intrinsic nuclear excitations described within the quasiparticle random-phase approximation (QRPA). Various kinds of SA corrections are treated at the same theoretical ground. The known corrections are well reproduced. As relevant cases, we consider subtraction of SA related with (i) violation of the translational invariance (isovector $E1$ and isoscalar toroidal and compression $E1$ modes), (ii) pairing-induced nonconservation of the particle number $\left(E2\right(K=0)$ and $E0$ modes), and (iii) rotational invariance $\left(E2\right(K=1)$ and $M1(K=1)$ modes). The SA subtraction can be done at the level of QRPA states, electromagnetic responses, and even transition operators. The additional deformation-induced corrections for $E1$ excitations are proposed and shown to be essential for the compression isoscalar mode. The accuracy of the method is demonstrated by Skyrme QRPA calculations for axially deformed ${}^{154}\mathrm{Sm}$.

Figure 1

The total QRPA photoabsorption cross section (black solid curve) and its $\mu =0$ and $\mu =1$ branches (green dashed and brown dashed-dotted curves) as compared with the experimental data [42].

Figure 2

The QRPA compression $E1$ strength function Eq. (41) in ${}^{154}\mathrm{Sm}$ calculated with the parametrization SLy6. The isoscalar (left panels) and isovector (right panels) strengths are considered. The branches $\mu =0$ (top), $\mu =1$ (bottom) are plotted for the polluted strength “no elim” (red dotted curves) and refined strengths “sph elim” and “full elim,” calculated without (blue dash curve) and with (black solid curves) deformation correction $d_q^{\mu }$, respectively.

Figure 3

The same as in Fig. 2 but for toroidal $E1$ response.

Figure 4

Skyrme QRPA strength function (41) for isoscalar (top panel) and isovector (bottom panel) $E0$ transitions. Results for $\mathrm{\Delta }T=0$ are compared with $(\alpha ,{\alpha }^{\prime })$ experimental data of Youngblood et al. [43] and M. Itoh et al. [44]. The strengths without (red dotted curve) and with (black solid curves) SA elimination are shown.

Figure 5

Isoscalar (left) and isovector (right) $E20$ ($\mu =0$) and $E21$ ($\mu =1$) strength functions calculated without (red dotted curves) and with (black solid curves) SA elimination.

Figure 6

The convective isoscalar current transition density in the x-z plane of the intrinsic frame, calculated following Eq. (33) for the first ($\nu =0$) spurious QRPA $K^{\pi }=1^{+}$ solution with the energy $\hslash {\omega }_0=0.95$ MeV. The panels show: (a) $\delta {\mathbf{j}}_{\nu =0}^{(\mathrm{\Delta }T=0)}\left(\mathbf{r}\right)$ without SA elimination, (b) the SA correction term in the right-hand side of Eq. (33), (c) $\delta {\mathbf{j}}_{{\nu }^{\prime }=0}^{(\mathrm{\Delta }T=0)}\left(\mathbf{r}\right)$ with SA elimination.

Figure 7

The total (top), orbital (middle), and spin (bottom) $M11$ strength functions calculated without (red dotted line) and with (black solid line) SA elimination.