The direct part of $\langle\Phi\vert\hat{\bm{P}}^2\vert\Phi\rangle$

Standard calculations for the SLy4 Skyrme functional [9] are performed for the Lipkin operator of $\hat{K}=\hat{T}/A$, which is a fixed factor of $A$ smaller than the one-body kinetic-energy operator $\hat{T}$. This procedure simply renormalizes the single particle masses as $m'=m(1-1/A)$ [15]. Another standard was adopted for the SLy6 and SLy7 functionals, whereby the Lipkin operator of Eq. (14) with the exact mass, that is $k=\hbar^2/2mA$, was used. In Fig. 5, are compared the Lipkin projected energies calculated with the exact and PY, Eq. (28), masses and Fig. 6 shows values of these masses. In all cases, the PY masses were calculated by using in Eq. (14) the shift of $\vert\bm{R}\vert=2$fm.

First one sees that the Lipkin projected energies obtained with the two-body Lipkin operators, Eq. (14), differ from those using the standard one-body operator $\hat{T}/A$ by up to 13MeV. The mass dependence of this difference can be very well described by the sum of the volume and surface terms. Therefore, its major part can easily be absorbed in the parameters of the Skyrme functional. This confirms conclusions of Ref. [16]. Second, one sees in Fig. 6 that the values of the PY masses differ form the exact ones only by a few percent. Therefore, the Lipkin projected energies based on these two prescriptions for the normalizing constant $k$ differ very little.

Figure 7: (Color online) Same as in Fig. 5 but for the chain of lead isotopes around $^{208}$Pb.

Results shown in Fig. 7 aim at checking whether the differences between the exact-mass and PY-mass correcting factors can influence shell effects in masses of lead isotopes. One sees, that these differences induce an almost constant shift of the energy, of the order of 0.5MeV. Moreover, the shell effect at $^{208}$Pb, induced by replacing the Lipkin operator $k\hat{\bm{P}}^2$ by $\hat{T}/A$, is very small.

Figure 8: (Color online) Mass-number dependence of the 0-th (lower panel) and 2-nd (upper panel) GOA moments of the two-body momentum operator squared and its direct part, see Eqs. (31) and (32).

Therefore, based on these results, one might be tempted to consider $\hat{T}/A$ as a viable alternative to $k\hat{\bm{P}}^2$. However, in view of the Lipkin symmetry-restoration method, this is not the case. This is shown in Fig. 8, where are compared the GOA properties of the reduced kernels of $\hat{\bm{P}}^2$ and $\hat{T}$,

$$p_2(\bm{R}) = \frac{\langle\Phi\vert{\hat{\bm{P}}^2}\vert\Phi(\bm{R})\rangle} ... ...i \vert\Phi(\bm{R})\rangle} = p_{20} -{\textstyle{\frac{1}{2}}}p_{22}\bm{R}^2 .$$ (31)

$${\pi}(\bm{R}) = \frac{\langle\Phi\vert{\textstyle{\frac{2m}{\hbar^2}}}{\hat{T}}\v... ...\vert\Phi(\bm{R})\rangle} = {\pi}_0 -{\textstyle{\frac{1}{2}}}{\pi}_2\bm{R}^2 ,$$ (32)

where the factor in front of the kinetic-energy operator was chosen so that $\pi$ is simply the kernel of the direct part of the operator $\hat{\bm{P}}^2$ [16]. Again, the quality of the GOA is here excellent, so the parameters $p_{22}$ and ${\pi}_2$ characterize the kernels very well, while the parameters $p_{20}=\langle\Phi\vert{\hat{\bm{P}}^2}\vert\Phi\rangle$ and ${\pi}_0=\langle\Phi\vert{\textstyle{\frac{2m}{\hbar^2}}}{\hat{T}}\vert\Phi\rangle$ give the standard average values.

In the bottom panel of Fig. 8, one sees that the direct parts (${\pi}_0$) increase almost linearly with the particle number $A$, while the true average values ($p_{20}$) increase more like $A^{3/4}$. In a heavy nucleus like $^{208}$Pb, these trends result in $p_{20}$ being overestimated by ${\pi}_0$ by about a factor of 3, which confirms the results obtained in Ref. [17]. Such mass dependencies explain why this error can be fairly well absorbed within the volume ($\sim$$A$) and surface ($\sim$$A^{2/3}$) energies of the Skyrme functional.

However, in the top panel of Fig. 8, one sees that the second moments of these kernels, $p_{22}$ and ${\pi}_2$, behave quite differently. The second moment of the true kernel ($p_{22}$) increases quite fast, like $A^{3/2}$, while the direct part (${\pi}_2$) only as $A^{1/2}$. In $^{208}$Pb, these trends result in $p_{22}$ being underestimated by ${\pi}_2$ by about a factor of 200. It is then obvious that one cannot replace the true average values $\langle\Phi\vert\hat{\bm{P}}^2\vert\Phi\rangle$ by the direct parts, no matter which value of the multiplicative factor is used.

One can note here in passing that, within the GOA, Eq. (29) gives the relation $p_{22}=\frac{2}{9}p_{20}^2$, which is very well fulfilled by the numerical results shown in Fig. 8. Than, again within GOA, the average values of $\hat{T}$ in the projected states can be calculated in analogy to Eq. (9), and are proportional to

$$\pi^{\mbox{\scriptsize {GOA}}}(\bm{P}) = \pi_0-\frac{3\pi_2}{2a}+\frac{\pi_2}{2a^2}\bm{P}^2 ,$$ (33)

The fact that $\pi_2$ is a factor of 200 too small in $^{208}$Pb means that the function $\pi(\bm{P})$ is not able to flatten the $\bm{P}^2$ dependence of $E_\Phi(\bm{P})$, and thus the method based on replacing $k\hat{\bm{P}}^2$ by $\hat{T}/A$ does not lead to a proper VAP estimate of the projected energy.