The isovector 0$^+$ and isoscalar, isovector 2$^+$ modes

Figure 6 displays strength functions for the 0$^+$ and 2$^+$ channels in $^{120}$Sn and $^{174}$Sn. (We discussed the isoscalar $0^+$ mode above to illustrate the accuracy of our solutions, but include it here as well for completeness.)

Figure 6: Isoscalar and isovector strength functions for (a) the 0$^+$ channel of $^{120}$Sn, (b) the 0$^+$ channel of $^{174}$Sn, (c) the 2$^+$ channel of $^{120}$Sn, and (d) the 2$^+$ channel of $^{174}$Sn. The cutoff $\epsilon_{\rm crit}$ is 150 MeV and $v^2_{\rm crit}$ is $10^{-12}$.

The calculations show the appearance of low-energy $0^+$ strength -- both isovector and isoscalar -- and low-energy isovector 2$^+$ strength in $^{174}$Sn, though in none of these instances is the phenomenon quite as dramatic as in the isoscalar $1^-$ channel.

The EWSR for the isoscalar 2$^+$ transition operator,

$$\hat{F}_{2M}= e{\frac{Z}{A}} \sum_{i=1}^A r_i^2 Y_{2M}(\Omega_i),$$ (8)

can be written as [75]

$$\sum_k \sum_M E_k\vert\langle k\vert\hat{F}_{2M}\vert\rangle... ...25}{4\pi}\frac{e^2\hbar^2}{m}\frac{Z^2}{A} \langle r^2\rangle.$$ (9)

The sum rule is obeyed as well in the 2$^+$ isoscalar channel as in the $0^+$ and $1^-$ channels, the only difference being that one needs to include quasiparticle states with $j > 15/2$ for $^{174}$Sn. For $^{120}$Sn ($^{174}$Sn) from Fig. 6, the EWSR is 37222 (34971) $e^2$MeVfm$^4$ while the QRPA value is 37030 (35010) $e^2$MeVfm$^4$.

While on the topic of the sum rule, we display in Table 2 the $j_{\rm max}$-dependence of the EWSR for several channels in $^{150}$Sn, with $R_{\rm box}= 25$ fm. By taking $j_{\rm max}= 19/2$ we appear to obtain essentially the entire strength in all three cases.

Table 2: The $j_{\rm max}$-dependence of isoscalar EWSR for $^{150}$Sn. $R_{\rm box}$ is 25 fm.

$\parbox{10cm}{ \begin{tabular}{lcrr} $T$~$J^\pi$\ & units &$j_{\rm ... ...\ IS 2$^+$\ & $(e^2~{\rm MeV}~{\rm fm}^4)$\ & 35542 & 35445 \\ \end{tabular}}$