Properties of isoscalar-pair condensates

It is pointed out that the ground state of $n$ neutrons and $n$ protons in a single-$j$ shell, interacting through an isoscalar ($T=0$) pairing force, is not paired, $J=0$, but rather spin aligned, $J=n$. This observation is explained in the context of a model of isoscalar $P$ ($J=1$) pairs, which is mapped onto a system of $p$ bosons, leading to an approximate analytic solution of the isoscalar-pairing limit in $jj$ coupling.

Figure 1

Energies (in units of the pairing strength $g$) of the lowest two $T=0$ states for $N=4$ particles in an $f_{7/2}$ shell as a function of the relative mixture $x$ of isovector and isoscalar pairing.

Figure 2

Energies (in units of the pairing strength $g$) of the lowest two $T=0$ states for $N=6$ particles in the $f_{7/2}$ shell as a function of the relative mixture $x$ of isovector and isoscalar pairing.

Figure 3

Energies (in units of the pairing strength $g$) of the $T=0$ ground state for $N=2$ particles (top) and of the lowest two $T=0$ states for $N=4$ particles (bottom) as a function of the spin-orbit splitting between the $f_{7/2}$ and $f_{5/2}$ orbits and for pure isoscalar pairing. The shaded area in the top panel indicates the critical value of the spin-orbit splitting, $v_{\ell s}^{*}$, at which the $jj$ coupling takes on. (See text for details.)