Isospin mixing

Figure 1: Isospin impurities in the ground state of $^{80}$Zr, predicted by DFT, using various Skyrme parametrizations [21] plotted versus the corresponding excitation energies of the $T=1$ doorway states. Open dots mark results obtained before the Coulomb rediagonalization (BR), $\alpha_C^{(BR)} = 1 - \vert b_{T=\vert T_z\vert,T_z}\vert^2$, which were calculated by using expansion coefficients of Eq. (1). Full dots mark the impurities (5) obtained after the Coulomb rediagonalization (AR).

By using the perturbation theory [22] and the analytically solvable hydrodynamical model [23], the isospin mixing in atomic nuclei has been studied since the 1960s (see Ref. [24] for a review). These simple approaches accounted for such qualitative features of the isospin impurities like the steady increase in $N=Z$ nuclei with increasing proton number and strong quenching with increasing $\vert N-Z\vert$. Quantitatively, however, their predictions for the values of the isospin impurities $\alpha_C$ were not very reliable.

Increased demand for accurate values of isospin mixing has been stimulated by the recent high-precision measurements of superallowed $\beta$-decay rates [6,7]. Large-scale shell-model approaches [25], although very accurate in the description of configuration mixing, can hardly account for the long-range polarization exerted on the neutron and proton states by the Coulomb force whose accurate treatment requires using large configuration spaces. In contrast, in self-consistent DFT, such polarization effects are naturally accounted for by finding the proper balance between the Coulomb force, which tends to make the proton and neutron states different, and the isoscalar part of the strong force, which has an opposite tendency.

In general, isospin impurities determined without removing spurious isospin mixing are underestimated by about 30% compared to the values obtained after rediagonalization [12]. In the particular case of $^{80}$Zr, the removal of spurious admixtures increases $\alpha_C$ from $\sim$2.9% to $\sim$4.4%, as illustrated in Fig. 1. It is encouraging to see that the latter value agrees well with the central value of empirical impurity deduced from the giant dipole resonance $\gamma$-decay studies, as communicated during this meeting by F. Camera et al. [26]. Unfortunately, experimental error bars are too large to discriminate between various Skyrme parametrizations, which differ in predicted values of $\alpha_C$ by as much as $\sim$10%.

Figure 2 illustrates our attempts to correlate the values of $\alpha_C$ with the surface and volume symmetry energies, which are primary quantities characterizing the isovector parts of nuclear EDFs. The linear regression coefficients shown in the figure hardly indicate any correlation of $\alpha_C$ with these quantities. In fact, no clear correlation was found between the calculated values of $\alpha_C$ and other bulk characteristics of the Skyrme EDFs, including the isovector and isoscalar effective masses, and incompressibility.

Figure 2: Isospin impurities predicted by several Skyrme EDFs for the ground states of $^{40}$Ca (left) and $^{100}$Sn (right) plotted versus the surface (top) and volume (bottom) symmetry energy.