Numerical applications: the isospin mixing and the isospin-projected energies in N=Z nuclei

The isospin-mixing parameter, calculated before and after rediagonalization, is defined as $\alpha_C = 1- \vert b_{\vert T_z\vert,T_z}\vert^2$ and $\alpha_C = 1- \vert a^{n=1}_{\vert T_z\vert,T_z}\vert^2$, respectively. Its theoretical accuracy depends on different factors, and in particular, on the size of the spherical harmonic-oscillator (HO) basis used in the calculations. A choice of the number of the HO shells $N_0$ included in such calculations is always a result of a trade-off between the accuracy and the CPU-time efficiency. In this respect, a bottle-neck in our calculation scheme is the exact treatment of the exchange Coulomb contribution, which makes calculations prohibitively time consuming for $N_0 > 16$.

Figure 1: The isospin-mixing parameter $\alpha _C$ calculated in $^{100}$Sn (after rediagonalization) as a function of the number of the HO shells $N_0$. The results were obtained by using the SIII[17] Skyrme parameterization.

Dependence of the isospin-mixing parameter on $N_0$ is depicted in Fig. 1. The figure shows $\alpha _C$ in $^{100}$Sn, calculated after rediagonalization, by using the SIII Skyrme parameterization of Ref.[17]. In the expanded scale of the figure, a significant variation of the mixing parameter with $N_0$ is clearly seen. Unfortunately, the mixing parameter $\alpha _C$ does not stabilize at $N_0=16$. Hence, by using the present method, $\alpha _C$ cannot be calculated with the absolute precision greater than $\pm 0.002$, or with the relative precision grater than $\pm 4$%. However, our studies show that the inaccuracy in evaluating $\alpha _C$ due to the basis cut-off appears to be much smaller than the uncertainty related to the Skyrme force parameterization[18].

Figure 2: Total binding energies in $^{16}$O, $^{40}$Ca, $^{56}$Ni, and $^{100}$Sn as a functions of the number of the HO shells $N_0$. Calculations were performed by using the SII Skyrme force[17]. Dots and squares label the binding energies calculated before and after rediagonalization, respectively.

Fig. 2 shows the total binding energies versus $N_0$, calculated for doubly magic nuclei: $^{16}$O, $^{40}$Ca, $^{56}$Ni, and $^{100}$Sn. This set of calculations was performed by using the SII Skyrme force parameterization[17]. The curves labeled by black squares depict the projected energies $E^{\textrm{BR}}_{T,T_z}$ (4), calculated before rediagonalization, and those marked by triangles show the total binding energies $E^{\textrm{AR}}_{n=1,T_z}$ (7), obtained after rediagonalization of the total Hamiltonian in the isospin-projected basis. The figure shows that (i) the Coulomb rediagonalization effect increases with increasing $Z$ as anticipated, and that (ii) the choice of $N_0=12$ HO shells provides a reasonable estimate for the total binding energy even for $^{100}$Sn. Hence, all calculations presented below are done for $N_0=12$.

Figure 3: The isospin mixing (left) and the difference between the total binding energy and the HF energy (right) in $N=Z$ nuclei, calculated for the SIII Skyrme force and $N_0=12$ HO shells. The results shown by open and full dots represent variants of the calculation before and after rediagonalization, respectively.

Fig. 3a shows the isospin mixing in $N=Z$ nuclei as a function of the mass number $A$. Results obtained before and after rediagonalization are shown by open and full dots, respectively. In both variants of the calculations, the isospin mixing shows a gradual increase as a function of $A$. It increases from a fraction of a percent in $^{16}$O to about 4%-5%, depending on the variant of the calculation. Note that the results obtained before rediagonalization follow closely those obtained in Ref.[19].

The isospin mixing obtained after removing the spurious mean-field component through the Coulomb rediagonalization is systematically larger than the one obtained within the HF method followed by the exact isospin projection. This result confirms that the mean-field breaks the isospin symmetry in such a way that it counterbalances the external symmetry breaking mechanism caused by the Coulomb field. Nevertheless, as clearly seen in Fig. 3b, the HF energy is astonishingly close to the total energy obtained after the Coulomb rediagonalization.