A=62 nuclei: $^{62}$Zn and $^{62}$Ga

For $^{62}$Zn, the results of the NCCI calculations of the low-lying $0^+$ states were communicated in Ref. [18]. Here, for the sake of completeness, we briefly summarize the results obtained therein. The calculated spectrum of the $0^+$ states below the excitation energy of 5MeV is shown in Fig. 9. The NCCI calculations were based on six reference states that include: the ground state, the two lowest neutron p-h excitations $\nu_1$ and $\nu_2$, the two lowest proton p-h excitations $\pi_1$ and $\pi_2$, and the lowest proton 2p-2h excitation $\pi\pi_1$. Their properties are listed in Table 9.

As discussed in Ref. [18], the calculated spectrum of $0^+$ states is in a very good agreement with the recent data communicated by Leach et al. [59]. As shown in Fig. 10(a), the calculated total g.s. energy is stable with increasing the number of reference configurations. Its value of $-$526.595MeV ($N=12$ harmonic oscillator shells were used) underestimates the experiment by roughly 2%.

Table 9: Similar as in Table 3, but for $^{62}$Zn. Here, the Slater determinants are labeled by neutron and proton configurations described in the text. The last column shows energies of the lowest $0^+$ states projected from a given Slater determinant.

$i$	$\vert^{62}$Zn$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$j_\nu$	$j_\pi$	$k$	$\Delta$E $_{\rm I=0}$
1	g.s.	$-$521.549	0.270	31$^\circ$	0.000	0.000		$-$526.405
2	$\pi_1$	1.433	0.286	20$^\circ$	0.005	0.152	Y	2.036
3	$\nu_1$	3.347	0.255	40$^\circ$	0.689	0.318	X	3.703
4	$\nu_2$	4.287	0.240	25$^\circ$	$-$0.281	$-$0.325	Y	3.852
5	$\pi_2$	5.251	0.246	48$^\circ$	$-$0.103	$-$0.076	X	5.672
6	$\pi\pi_1$	3.381	0.251	38$^\circ$	0.000	0.000		3.471

Table 10: Same as in Table 9, but for $^{62}$Ga.

$i$	$\vert^{62}$Ga$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$j_\nu$	$j_\pi$	$k$	$\Delta$E $_{\rm I=0}$
1	Y	$-$512.122	0.268	30$^\circ$	$-$0.138	0.149	Y	$-$516.930
2	X	0.007	0.268	30$^\circ$	0.180	$-$0.170	X	$-$0.001
3	Z	0.190	0.269	30$^\circ$	$-$0.299	0.264	Z	0.005
4	$\pi_1$	1.266	0.284	20$^\circ$	$-$0.012	$-$0.264	X	2.175
5	$\nu_1$	1.977	0.255	35$^\circ$	$-$0.440	$-$0.351	X	3.151

Figure 9: The low-lying $0^+$ states in $^{62}$Zn. The first two columns show old and new experimental data, see [59] for details. The next three columns collect the results of the shell-model calculations using interactions MSDI3 [60], GXPF1 [61], and GXPF1A [62], respectively. The last three columns show results obtained within the NCCI approach, angular-momentum projection, and pure HF method, respectively. From Ref. [18].

In spite of the fact that the total binding energy is relatively stable, the calculated ISB corrections to superallowed transition $^{62}$Ga $\rightarrow ^{62}$Zn strongly depend on the details of the calculation. This is illustrated in Fig.10(b), which shows values of $\delta _{\rm C}$ in function of the number of configurations taken for the NCCI calculations in the daughter nucleus $^{62}$Zn. The four different curves correspond to different model spaces taken for the NCCI calculation in the parent nucleus $^{62}$Ga, see Table 10 and Fig. 10(c). In terms of Nilsson numbers, counted relatively to the $^{64}$Zn$_{32}$ even-even core, the configurations X,Y,Z correspond to differently aligned $\nu \vert 312\, 3/2\rangle^{-1}\otimes \pi \vert 312\, 3/2\rangle^{-1}$ two-hole states, $\pi_1$ denotes $\nu \vert 312\, 3/2\rangle^{-1}\otimes \pi \vert 312\, 5/2\rangle^{-1}$, two hole state while $\nu_1$ is $\nu \vert 321\, 1/2\rangle^{-1}\otimes \pi \vert 312\, 3/2\rangle^{-1}$. The three curves labeled with open dots, and open and filled triangles correspond to states $0^+$ projected from the [X,Y], [X,Y,Z], and [X,Y,Z,$\pi_1$] configurations, respectively. These curves essentially overlap with each other, thus showing no influence of the configuration-mixing (in this restricted model space) on the structure of the $0^+$ state in the parent nucleus. Note, however, that an extension of the model space by adding the lowest neutron p-h excitation, [X,Y,Z,$\pi_1$,$\nu_1$], leads to an increase in $\delta _{\rm C}$ of about 1%. Note also, that all curves are particularly sensitive to an admixture of the $\nu_2$ configuration in the daughter nucleus. This admixture increases $\delta _{\rm C}$ by almost 4%. The analysis clearly shows that, within the present implementation of the model, it is essentially impossible to match the spaces of states used to calculate the parent and daughter nuclei. The reasons are manifold. The lack of representability of the $T=1,I=0$ states in the $N=Z$ nucleus within the conventional MF using products of neutron and proton wave functions and difficulties in constraining the time-odd part of the functional are two of them. Difficulty of matching the model spaces in the parent and daughter nuclei introduce here an artificial ISB effect. As a result, beyond a simple mixing of orientations used in the result given in Table 1, the NCCI approach cannot be used for determining the ISB corrections to the transition $^{62}$Ga $\rightarrow ^{62}$Zn.

Figure 10: (a) The low-lying $0^+$ states in $^{62}$Zn in function of the number of configurations included in the NCCI calculations. (b) Calculated ISB corrections versus the number of configurations taken into account in the daughter nucleus. Different curves correspond to different sets of configurations taken to calculate $0^+$ state in $^{62}$Ga. (c) The HF energies of configurations included in the calculation of $^{62}$Ga, see Table 10. Further details are given in the text.