No-core configuration-interaction calculations for $0^+$ states in $^{62}$Zn

A large difference between the ISB corrections to the $^{62}$Ga$\rightarrow$$^{62}$Zn Fermi matrix element, calculated using the DFT and SM+WS approaches, is one of the motivations to undertake the NCCI studies of the participating nuclei. Interestingly, nucleus $^{62}$Zn has been recently remeasured by the TRIUMPH group [6], and its spectrum of low-lying $0^+$ states is now posing a great challenge to theory, as shown in the Tab. I and Fig. 1. Both the table and figure also include results of our NCCI study, which involves the mixing of $0^+$ states projected from six reference states. They comprise the deformed ground state (g.s.) and five low-lying excited HF configurations, including two lowest proton ($\pi_1$ and $\pi_2$) and two lowest neutron ($\nu_1$ and $\nu_2$) p-h excitations, and the lowest proton-proton 2p-2h configuration ($\pi\pi_1$). The excited states are self-consistent p-h excitations with respect to the g.s. configuration $\nu[7,7,9,9]\pi[7,7,8,8]$, where the labels denote the numbers of neutrons and protons occupying the lowest Nilsson levels in each parity-signature block, $(\pi, r) = (+,+i),(+,-i),(-,+i),(-,-i)$. Their configurations expressed in terms of the Nilsson quantum numbers $[N,n_z,\Lambda,\Omega]$, corresponding to the dominant components of the particle and hole orbits, are given in Fig. 1.

Table: Excitation energies of $0^+$ states in $^{62}$Zn up to 5MeV. First two columns show old and new experimental data, see [6] for details. Next three columns collect the results of shell-model calculations using MSDI3 [13], GXPF1 [14], and GXPF1A [15] interactions, respectively. Last column shows the results obtained in this work using six reference Slater determinants described in the text. All values are in keV.

OLD	NEW	MSDI3	GXPF1	GXPF1A	SV$^{{\rm mix}}$
2341.95(2)		2263	2320	2094
		2874		2811
3042.9(8)	3045.5(4)	3071		3457	2953
	3862(2)	3513	3706	3682	3884
4008.4(7)	3936(6)	3833		3991	4263
				4444
4620(20)	4552(9)	4551	4729	4643	4347

Figure: The first five columns show low-energy $I$=0$^+$ states in $^{62}$Zn listed in Table I. Columns marked HF and SV$^{{\rm proj}}$ show HF results and those for the $0^+$ states projected from the HF configurations before the mixing, respectively.

Figure: Stability of the absolute energies of the calculated $0^+$ states in $^{62}$Zn with respect to number of reference configurations included in the calculations.

Figure: Calculated energies of the $0_2^+$ relative to $0_1^+$ states in $^{38}$Ca and $^{38}$K, and the ISB corrections to the corresponding $0_1^+\rightarrow 0_1^+$ and $0_2^+\rightarrow 0_2^+$ transitions. Empirical excitation energy of the $0_2^+$ state in $^{38}$Ca is also shown.

It is gratifying to observe that our model is able to capture, without adjusting a single parameter, the spectrum of $0^+$ states in $^{62}$Zn very accurately, even better than the state-of-the-art SM calculations. Moreover, as shown in Fig. 2, the calculated spectrum of the $0^+$ states in $^{62}$Zn is relatively stable with respect to increasing the number of reference configurations. The last two columns of Fig. 1 illustrate the importance of symmetry restoration and configurations mixing.

Unfortunately, the calculated corrections $\delta_{\rm C}$ are sensitive to tiny admixtures to the wave function and, and present, the calculated values are not stable. For example, by adding the $0^+$ state projected from configuration $\nu_2$, one changes the absolute g.s. energy of $^{62}$Zn by only $\approx$200keV, but at the same time $\delta_{\rm C}$ changes by $\approx$4%. Such a large change of $\delta_{\rm C}$ is probably entirely artificial, reflecting the fact that the spaces of states used to calculate the parent and daughter nuclei do not match.