---
abstract: '[Starting from a Skyrme interaction with tensor terms, the $\beta$-decay rates of $^{52}$Ca have been studied within a microscopic model including the $2p-2h$ configuration effects. We observe a redistribution of the strength of Gamow-Teller transitions due to the $2p-2h$ fragmentation. Taking into account this effect results in a satisfactory description of the neutron emission probability of the $\beta$-decay in $^{52}$Ca.]{}'
author:
- ' $^{1),2)}$'
title: 'Strength fragmentation of Gamow-Teller transitions and delayed neutron emission of atomic nuclei'
---

The multi-neutron emission is basically a multistep process consisting of (a) the $\beta$-decay of the parent nucleus (N, Z) which results in feeding the excited states of the daughter nucleus (N - 1, Z + 1) followed by the (b) $\gamma$-deexcitation to the ground state or (c) multi-neutron emissions to the ground state of the final nucleus (N - 1 - X, Z + 1), see e.g., Ref. [@b05]. Predictions of the multi-neutron emission are needed for the analysis of radioactive beam experiments and for modeling of astrophysical r-process. Recent experiments gave an evidence for strong shell effects in exotic calcium isotopes [@w13; @s13]. For this reason, the $\beta$-decay properties of neutron-rich isotope $^{52}$Ca provides valuable information [@h85], with important tests of theoretical calculations.

[ccccccc]{} $\lambda_i^{\pi}=1_i^+$&&\
& Expt. & QRPA & 2PH & Expt. & QRPA & 2PH\
$1_1^+$ &1.64 & 1.5 & 1.3 &4.2$\pm$0.1& 4.3 & 4.3\
$1_2^+$ &2.75 & & 3.9 &4.5$\pm$0.2& & 6.4\
$1_3^+$ &3.46 & & 4.2 &5.3$\pm$0.5& & 9.2\
$1_4^+$ &4.27 & 5.0 & 4.9 &4.0$\pm$0.5& 3.2 & 3.3\

One of the successful tools for studying charge-exchange nuclear modes is the quasiparticle random phase approximation (QRPA) with the self-consistent mean-field derived from a Skyrme energy-density functional (EDF) since these QRPA calculations enable one to describe the properties of the parent ground state and Gamow-Teller (GT) transitions using the same EDF. Making use of the finite rank separable approximation (FRSA) [@gsv98] for the residual interaction, the approach has been generalized for the coupling between one- and two-phonon components of the wave functions [@svg04]. The FRSA in the cases of the charge-exchange excitations and the $\beta$-decay was already introduced in Refs. [@svg12; @ss13] and in Ref. [@svbag14; @e15], respectively. In the case of the $\beta$ decay of $^{52}$Ca, we use the EDF T45 which takes into account the tensor force added with refitting the parameters of the central interaction [@TIJ]. The pairing correlations are generated by a zero-range volume force with a strength of -315 MeVfm$^{3}$ and a smooth cut-off at 10 MeV above the Fermi energies [@svbag14]. This value of the pairing strength has been fitted to reproduce the experimental neutron pairing energy of $^{52}$Ca obtained from binding energies of neighbouring Ca isotopes.

Taking into account the basic ideas of the quasiparticle-phonon model (QPM) [@solo; @ks84], the Hamiltonian is then diagonalized in a space spanned by states composed of one and two QRPA phonons [@svbag14], $$\begin{aligned}
\Psi _\nu (J M) = \left(\sum_iR_i(J \nu )Q_{J M i}^{+}+
\sum_{\lambda _1i_1\lambda _2i_2}P_{\lambda _2i_2}^{\lambda
_1i_1}( J \nu )\left[ Q_{\lambda _1\mu _1i_1}^{+}\bar{Q}_{\lambda
_2\mu _2i_2}^{+}\right] _{J M }\right)|0\rangle, \label{wf}\end{aligned}$$ where $Q_{\lambda \mu i}^{+}\mid0\rangle$ are the wave functions of the one-phonon states of the daughter nucleus (N - 1, Z + 1); $\bar{Q}_{\lambda\mu i}^{+} |0\rangle$ is the one-phonon excitation of the parent nucleus (N, Z). We use only the two-phonon configurations $[1^{+}_{i}\otimes 2^{+}_{i'}]_{QRPA}$. In the allowed GT approximation, the $\beta^{-}$-decay rate is expressed by summing the probabilities (in units of $G_{A}^{2}/4\pi$) of the energetically allowed transitions ($E_{k}^{\mathrm{GT}}\leq Q_{\beta}$) weighted with the integrated Fermi function $$\begin{aligned}
T_{1/2}^{-1}=D^{-1}\left(\frac{G_{A}}{G_{V}}\right)^{2}
\sum\limits_{k}f_{0}(Z+1,A,E_{k}^{\mathrm{GT}})B(GT)_{k},\end{aligned}$$ $$E_{k}^{\mathrm{GT}}=Q_{\beta}-E_{1^+_k},$$ where $G_A/G_V$=1.25 and $D$=6147 s. $E_{1_k^+}$ denotes the excitation energy of the daughter nucleus. As proposed in Ref. [@ebnds99], this energy can be estimated by the following expression: $$E_{1^{+}_{k}}\approx E_{k}-E_{\textrm{2QP},\textrm{lowest}}.$$ $E_{k}$ are the eigenvalues of the wave functions (\[wf\]) and $E_{\textrm{2QP},\textrm{lowest}}$ corresponds the lowest two-quasiparticle energy. The difference in the characteristic time scales of the $\beta$ decay and subsequent particle emission processes justifies an assumption of their statistical independence (see Ref. [@b05] for more details). The $P_{n}$ probability of the delayed neutron emission is defined as the ratio of the integral $\beta$-strength to the excited states above the neutron separation energy of the daughter nucleus.

The spectrum of four low-energy $1^+$ states of $^{52}$Sc is shown in Table 1. The structure peculiarities are reflected in the $\log ft$ values. We find that the dominant contribution in the wave function of the first (fourth) $1^+$ state comes from the configuration $\{\pi1f_{7/2}\nu1f_{5/2}\}$ ($\{\pi1f_{7/2}\nu1f_{7/2}\}$). The inclusion of the four-quasiparticle configurations $\{\pi1f_{7/2}\nu1f_{5/2} \nu2p_{3/2}\nu2p_{1/2}\}$ and $\{\pi1f_{7/2}\nu1f_{5/2} \nu2p_{3/2}\nu2p_{3/2}\}$ plays the key role in our calculations of the states $1_{2}^+$ and $1_{3}^+$, respectively. The inclusion of the two-phonon configurations results in the $P_{n}$ value of 5%, and the quantitative agreement with the experimental data [@h85] is satisfactory. Note that this value is almost three times less than that within the one-phonon approximation.

In summary, by starting from the Skyrme mean-field calculations the GT strength in the $Q_{\beta}$-window has been studied within the model including the $2p-2h$ fragmentation. We analyze this effect on the $\beta$-transition rates in the case of $^{52}$Ca. Including the $2p-2h$ configurations leads to qualitative agreement with existence of four low-energy $1^+$ states of $^{52}$Sc. As a result, the probability of the delayed neutron emission is decreased.

I would like to thank I.N. Borzov, Yu.E. Penionzhkevich, and D. Verney for fruitful collaboration, N.N. Arsenyev and E.O. Sushenok for help. This work is partly supported by CNRS-RFBR Agreement No. 16-52-150003, the IN2P3-JINR agreement, and RFBR Grant No. 16-02-00228.

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