Closed expression for the pair vibrational correlation energy of a uniform distribution of single-nucleon levels

A closed expression is derived for the pair vibrational correlation energy generated in the random-phase approximation by the isovector pairing force in the case when Kramers and charge-degenerate single-nucleon levels are uniformly distributed in an interval. The expression is used to analyze the spectral density of pair vibrational frequencies relative to that of two-quasinucleon energies. Applications to the analysis of the symmetry energy of the isovector pairing model and to a Strutinskij renormalization of this model are discussed.

Figure 1

The function $I(a,0)$. This function gives the two-neutron or two-proton or for $N=Z$ the neutron-proton pair vibrational correlation energy in units of the gap parameter ${\mathrm{\Delta }}_{\tau }$. The argument $a$ is the reciprocal coupling constant $G$ in units of the single-nucleon level spacing $1/g$; see Eq. (12).

Figure 2

The function $I(a,0)/sinha$. This function is proportional to the two-neutron or two-proton or for $N=Z$ the neutron-proton pair vibrational correlation energy for a constant single-nucleon level spacing $1/g$ and pair coupling constant $G$. As to the argument $a$ see the caption to Fig. 1.

Figure 3

The function $\delta I(a,l)$ for several $a$. This function gives the increment with a neutron excess of the neutron-proton pair vibational correlation energy in units of the gap parameter ${\mathrm{\Delta }}_n={\mathrm{\Delta }}_p$. The argument $l$ is the difference $\delta \lambda ={\lambda }_n-{\lambda }_p$ of the neutron and proton chemical potentials in units of twice the gap parameter; see Eq. (50). As to the parameter $a$ see the caption to Fig. 1. The dashed line indicates the approximation $\delta I(a,l)=l$.

Figure 4

The function $\Im h(z-i{10}^{-6})$ for $a,l=2,1$. This function gives (when ${10}^{-6}$ is viewed as infinitesimal) the cumulated density of the neutron-proton and for $l=0$ the two-neutron and two-proton pair vibrational frequencies relative to that of the two-quasinucleon energies. The argument $z$ is the vibrational frequency $\omega$ in units of twice the gap parameter ${\mathrm{\Delta }}_n={\mathrm{\Delta }}_p$. The labels on the abscissa axis and the dashed line are explained in the text.

Figure 5

The residual neutron-proton pair vibrational correlation energy $E_{\text{RPA},np}^{\text{res}}$ as a function of $T/A$. The dashed lines show least square fits by quadratic functions in the interval of the plot.

Figure 6

RPA energies of doubly even nuclei as functions of $A$. The solid lines shows the RPA energies from Ref. [21] and the dashed curves below each solid line are their smooth counterterms given by Eq. (51).