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The $\beta$-decay strength function $S_{\beta}(\textit{E})$ governs [1,2] the nuclear energy $\textit{E}$ distribution of elementary charge-exchange excitations and their combinations like proton particle $({\pi}p)$-neutron hole $({\nu}h)$ coupled into a spin-parity $I^{\pi}$: $[{\pi}p \otimes {\nu}h]I^{\pi}$ and neutron particle $({\nu}p)$-proton hole $({\pi}h)$ coupled into a spin-parity $I^{\pi}: [{\nu}p \otimes {\pi}h]I^{\pi}$. The strength function of Fermi-type $\beta$-transitions takes into account excitations $[{\pi}p \otimes {\nu}h]0^{+}$ or $[{\nu}p \otimes {\pi}h]0^{+}$. Since isospin is a quite good quantum number, the strength of the Fermi-type transitions is concentrated in the region of the isobar-analogue resonance ($IAR$). The strength function for $\beta$-transitions of the Gamow–Teller ($GT$) type describes excitations $[{\pi}p \otimes {\nu}h]1^{+}$ or $[{\nu}p \otimes {\pi}h]1^{+}$. At excitation energies $E$ smaller than $ \textit{Q}_{\beta} $ (total $\beta$-decay energy), $S_{\beta}(\textit{E})$ determines the characters of the $\beta$-decay. For higher excitation energies that cannot be reached with the $\beta$-decay, $S_{\beta}(\textit{E})$ determines the charge exchange nuclear reaction cross sections, which depend on the nuclear matrix elements of the $\beta$-decay type.
Successful applications of the total absorption $\gamma$-spectroscopy ($TAGS$) for $S_{\beta}(E)$ resonance structure study, methods of $TAGS$ spectra interpretation, and results of analysis of $S_{\beta}(E)$ structure for the $GT$ $\beta^{+}/EC$ and $GT$ $\beta^{-}$-decays were summarized in [1]. Development of experimental technique allows application of methods of nuclear spectroscopy with high energy resolution for $S_{\beta}(\textit{E})$ fine structure measurement [2-4]. First results of the $S_{\beta}(E)$ fine structure study were summarized in [2]. The combination of the $TAGS$ with high resolution $\gamma$-spectroscopy may be applied for detailed decay schemes construction [2]. It was shown [2-5] that the high-resolution nuclear spectroscopy methods give conclusive evidence of the resonance structure of $S_{\beta}(\textit{E})$ for $GT$ and first-forbidden ($FF$) $\beta$-transitions in spherical, deformed, and transition nuclei. High-resolution nuclear spectroscopy methods [2-4] made it possible to demonstrate experimentally the reveal splitting of the peak in the $S_{\beta}(\textit{E})$ for the $GT$ $\beta^{+}/EC$-decay of the deformed nuclei into two components.
Resonance structure of the $S_{\beta}(\textit{E})$ for $\beta$-decay of halo nuclei was analyzed in [6-8]. It was shown that when the parent nucleus has $\textit{nn}$ Borromean halo structure, then after $GT$ $\beta^{-}$ - decay of parent state or after $M1$ $\gamma$-decay of $IAR$ the states with $\textit{np}$ tango halo structure or mixed $\textit{np}$ tango + $\textit{nn}$ Borromean halo structure can be populated.
In this report the fine structure of $S_{\beta}(\textit{E})$ is analysed. Resonance structure of $S_{\beta}(\textit{E})$ for $GT$ and $FF$ $\beta$ – decays, structure of $S_{\beta}(\textit{E})$ for halo nuclei, quenching of the weak axial-vector constant ${{g_{A}}^{eff}}$, and splitting of the peaks in $S_{\beta}(\textit{E})$ for deformed nuclei connected with the anisotropy of oscillations of proton holes against neutrons (peaks in $S_{\beta}(\textit{E})$ of $GT$ $\beta^{+}/EC$–decay) or of protons against neutron holes (peaks in $S_{\beta}(\textit{E})$ of $GT$ $\beta^{-}$ – decay) are discussed.
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