The 0$^+$ isoscalar mode

In addition to the spurious state associated with nonconservation of particle-number by the HFB, the $0^+$ channel contains the important ``breathing mode''. In Table 1 we display results from a run with $v^2_{\rm crit}$=$10^{-12}$ for neutrons and $\varepsilon_{\rm crit}=150$ MeV for protons, resulting in the inclusion of 310 proton quasiparticle states and the same number of neutron states, with angular momentum up to $j=21/2$. The Table shows the QRPA energies and transition matrix elements of the particle-number operator. The spurious state is below 200 keV, well separated from the other states, all of which have negligible ``number-strength''. The nonzero number strength in the spurious state, like the nonzero energy of that state, is a measure of numerical error. If the space of two-quasiparticle states is smaller, with $\varepsilon_{\rm crit}=100$ MeV and $v^2_{\rm crit} = 10^{-8}$, the energy of the spurious state and the number strength barely change.

Table 1: The lowest-energy excited $0^+$ states in $^{174}$Sn. The second column shows the excitation energies and the third column the squared matrix elements of the particle-number operator between the $k$th excited state and the ground state ($k$=0).

$\parbox{7cm}{ \begin{tabular}{lll} {\it k} &$E_k$\ (MeV) & $\vert\la... ...0 & $0.252\times 10^{-5}$\ \\ 5 & 3.878 & $0.480\times 10^{-5}$ \end{tabular}}$

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

Figure 2 shows the strength function $S_J(E)$ for the isoscalar $0^+$ transition operator, cf. [75],

$$\hat{F}_{00} = \frac{eZ}{A}\sum_{i=1}^A r_i^2.$$ (3)

We have plotted three curves with successively more quasiparticle levels (from 246 proton levels and 203 neutron levels to 341 proton levels and 374 neutron levels), with cutoff parameters given in the figure caption. The major structures in the strength function are stable. The error remaining after to the gentlest truncation is extremely small.

The dependence of the strength function on the box size and quasiparticle cutoff is shown in Fig. 3. The upper part of the Figure (panels a-c) corresponds to a constant smoothing width of $\gamma$=0.5 MeV. This relatively small value is not sufficient to eliminate the finite-box effects but it allows us to assess the stability of the QRPA solutions as a function of $R_{\rm box}$. The large structure corresponding to the giant monopole resonance (GMR) is independent of box size no matter what the cutoff, but increasing the number of configurations magnifies the dependence on box size of local fluctuations in $S_J(E)$. The lower part of the Figure (panels d-f) are smoothed more realistically, as in Eq. (2). It is gratifying to see that the resulting strength functions are practically identical, i.e., the remaining dependence on $R_{\rm box}$ and the cutoff is very weak.

Figure 3: Isoscalar $0^+$ strength function in $^{174}$Sn for the box radii: $R_{\rm box}$=20fm (solid line) and $R_{\rm box}$=25fm (dotted line). In (a), (b), and (c) the smoothing-width parameter $\gamma$ is 0.5 MeV for all energies, while 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).

The energy-weighted sum rule (EWSR) for the isoscalar $0^+$ mode [75] is given by

$$\sum_k E_k\vert\langle k\vert\hat{F}_{00}\vert\rangle\vert^2 = 2\frac{e^2\hbar^2}{m}\frac{Z^2}{A}\langle r^2 \rangle ,$$ (4)

where the expectation value is evaluated in the HFB ground state. This sum rule provides a stringent test of self consistency in the QRPA. In $^{174}$Sn, the right-hand side of Eq. (4) is 35215 $e^2$ MeV fm$^4$ and the left-hand side 34985$\pm 15$ $e^2$ MeV fm$^4$ for all of the calculations of Fig. 3; the QRPA strength essentially exhausts the sum rule. (The QRPA values of the EWSR in this paper are obtained by summing up to $E_k=50$ MeV. )