The isoscalar 1$^-$ mode

The $1^-$ channel, home of the giant dipole resonance, the isoscalar squeezing resonance, and as yet incompletely understood low-energy peaks in neutron-rich nuclei (sometimes associated with skin excitations), has a spurious isoscalar mode associated with center-of-mass motion that can seriously compromise the low-energy spectrum if not handled with extreme care. We test the ability of our QRPA to do so in $^{100}$Sn, $^{120}$Sn, $^{174}$Sn, and $^{176}$Sn. (The nuclei $^{100}$Sn and $^{176}$Sn are the two-proton and two-neutron drip-line systems predicted by the HFB calculation with SkM$^{\ast}$. Neither nucleus has any static pairing, i.e., $\Delta_{\rm n}$= $\Delta_{\rm p}$=0.) In the following calculations, we take $\varepsilon_{\rm crit} = 140$ MeV for the protons and $v^2_{\rm crit}=9\times 10^{-12}$ for the neutrons. As discussed above, smoothed strength functions are practically independent of small changes in the cutoff. They are also independent of the cutoff in quasiparticle angular momentum provided we include all states with $j$$\leq$15/2.

Figure 4 shows the predicted isoscalar dipole strength function for $^{100,120,174,176}$Sn.

Figure: Isoscalar $1^-$ strength function in $^{100,120,174,176}$Sn for the corrected dipole operator in Eq. (6) (solid line) and the uncorrected operator in Eq. (5) (dotted line). The cutoff $\varepsilon_{\rm crit}$ is 140 MeV and $v^2_{\rm crit}$ 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.

For the transition operator, we use

$$\hat{F}_{1M}=\frac{eZ}{A}\sum_{i=1}^A r_i^3Y_{1M}(\Omega_i)~,$$ (5)

and the corrected operator

$$\hat{F}_{1M}^{\rm cor}=\frac{eZ}{A}\sum_{i=1}^A (r_i^3-\eta ... ...Y_{1M}(\Omega_i), \ \ \eta = \frac{5}{3} \langle r^2 \rangle ,$$ (6)

to remove as completely as possible residual pieces of the spurious state from the physical states [10]. The fact that the strength functions produced by these two operators -- displayed in Fig. 4 -- coincide so closely shows the extreme accuracy of our QRPA solutions; they are uncontaminated by spurious motion even without the operator correction. The spurious-state energies $E_{\rm spurious}$ are 0.964 MeV for $^{100}$Sn and 0.713 MeV for $^{120}$Sn, and the energies of the first physical excited states are 7.958 MeV for $^{100}$Sn and 7.729 MeV for $^{120}$Sn. In $^{174}$Sn ($^{176}$Sn), $E_{\rm spurious}$ is 0.319 MeV (0.349 MeV) and the first physical state is at 3.485 MeV (2.710 MeV), lower than in the more stable isotopes. Pairing correlations do not affect accuracy; the neutrons in $^{120}$Sn and $^{174}$Sn are paired, while those in $^{100}$Sn and $^{176}$Sn are not.

We display the fine structure of the isoscalar $1^-$ strength functions in $^{120}$Sn and $^{174}$Sn in Fig. 5, which also illustrates the dependence of the results on $R_{\rm box}$. The dependence is consistent with that of Fig. 3 for the isoscalar 0$^+$ strength; the low-amplitude fluctuations in $S_J(E)$ that are unstable as a function of $R_{\rm box}$ disappear, and the smoothed strength function depends only weakly on $R_{\rm box}$. In $^{120}$Sn, the two sharp peaks below 10 MeV correspond to discrete states while the broad maxima centered around 15 MeV and 27 MeV are in the continuum, well above neutron-emission threshold. A similar three-peaked structure emerges in $^{174}$Sn, though most of the strength there is concentrated in the low-energy peak at $E \approx 4$ MeV. Fig. 4 shows (as we will discuss in our forthcoming paper [76]) that the appearance of the low-energy isoscalar dipole strength is a real and dramatic feature of neutron-rich dripline nuclei [77,78].

Figure 5: Isoscalar $1^-$ strength function in $^{120}$Sn (left) and $^{174}$Sn (right) for two box radii: $R_{\rm box}$=20fm (solid line) and $R_{\rm box}$=25fm (dotted line). In (a) and (b) the smoothing-width parameter is constant ($\gamma$=0.5 MeV), while in (c) and (d) $\gamma (E)$ is given by Eq. (2).

The EWSR for the isoscalar $1^-$ mode [75] is

$$\sum_k \sum_M E_k\vert\langle k\vert\hat{F}_{1M}^{\rm cor}\v... ...1\langle r^4\rangle - \frac{25}{3}\langle r^2\rangle^2\right).$$ (7)

In $^{174}$Sn, the right-hand side is 403310 $e^2$ MeV fm$^6$, while the left-hand side is 400200 $e^2$ MeV fm$^6$. For $^{176}$Sn, the corresponding numbers are 406576 $e^2$ MeV fm$^6$ and 407100 $e^2$ MeV fm$^6$. This level of agreement is very good.