Gamow-Teller strength distributions of ¹⁸O and well-deformed nuclei ^24,26Mg by deformed QRPA

The Gamow-Teller (GT) excitation is one of the key transitions when studying the weak interaction of nuclei through charge-exchange (CEX) (or charged-current (CC)) reactions, beta decays, and electron or muon capture reactions. Recent remarkable progresses in the study of supernova (SN) neutrinos reveal that the GT transition is a key transition in the neutrino-induced reactions relevant to the neutrino processes Taka2015 ; Bala2015 ; Taka2013 ; Cheoun2012 ; Moha2023 ; Suzuki2023 ; Cheoun2023 . For example, the GT transition turns out to be dominant among various multipole transitions in the neutrino-nucleus ($\nu$-A) reaction, according to recent theoretical calculations Taka2015 ; Bala2015 . However, we do not have enough data to confirm the reliability of theoretical models. Only limited data for the neutrino-induced reactions are available until now, despite many discussions about the possibility of low-energy neutrino sources Sato2005 ; Shin2016 . The recent progress in JSNS² experiments JSNS2 could provide valuable low-energy neutrino beams on nuclear targets.

In this respect, neutrino reaction data at LSND LSND1 ; LSND2 , and recent successful CEX reaction data by proton or neutron beam at RIKEN or NSCL Sasano2011 ; Sasano2012 , triton and ³He beam at RCNP Freke2016 , and other light nuclei beams at other facilities, are greatly helpful for understanding the neutrino-induced reactions in the following aspects. First, the main contribution to the neutrino-induced reaction, namely, the GT transition, can be studied experimentally by the CEX reaction. Second, the quenching factor can be deduced from the GT data. Here, we should note that recent ab initio calculations have explained the quenching factor in a number of light- and medium-mass nuclei by the combined effect of two-body currents as well as by strong correlations Gysb2019 ; Heiko2020 . Third, some high-lying GT excitations, which contribute to the neutrino-induced reactions, can be investigated by the CEX reactions. These CEX data may also provide important information for the $r$-process nucleosynthesis study, because additional protons and/or neutrons in cosmological nucleosynthesis sites may change the nuclear species by means of CEX reactions. For example, Gamow-Teller (+) transitions from ²⁴Mg affect the electron capture rates in the O+Ne+Mg cores of stars, but theoretical results in Refs. Full1982 ; Jame2007 show some inconsistency among them. The understanding of decay branching ratios of the excited ²⁴Mg states are also important for ¹²C + ¹²C fusion reactions Muns2017 ; Tumi2018 ; Monp2022 . The GT results for ¹⁸O are an important input for astrophysical calculations such as the CNO cycle Moha2023 , and also for neutrino detection due to the admixture of ¹⁸O in natural water at Hyper-Kamiokande Abe2021 .

From the nuclear physics viewpoint, the GT transition is an allowed spin-isospin excitation of a nucleus, and the transition operator is one of the simplest. Consequently, it may give invaluable information of the nuclear spin-isospin structure. However, there still remain some ambiguities, that prevent us from having a full quantitative understanding of GT excitations throughout the nuclear chart; the same issues prevent us from knowing precisely the matrix elements for 2$\nu\beta\beta$ and 0$\nu\beta\beta$ decays. In this work, we focus on one of the ambiguities, or open questions, regarding the GT strength in deformed nuclei: we study the deformation effects on the GT strength, by using Deformed Quasi-particle Random Phase Approximation (DQRPA) calculations.

Many nuclei in the nuclear chart are considered to be deformed, which can be verified from the E2 transition data as well as the rotational band structure. In particular, neutron-rich and neutron-deficient nuclei may be largely deformed, and these nuclei are among those that have strong low-energy GT strength or short $\beta$-decay half-lives. Moreover, the deformation may produce unusual phenomena of shell evolution of protons and neutrons Ichik2019 , which make new sub-magic numbers and change to some extent the nature of particle-particle ($p-p$) and particle-hole ($p-h$) interactions. One of the practical but successful ways for properly describing the deformation is to start from the Nilsson model with the axial symmetry. Our previous calculations Ha2015a ; Ha2015b ; Ha2016 for the GT strength distribution of open shell nuclei by deformed QRPA showed that the deformation markedly alters the GT strength peaks obtained by a spherical QRPA.

The $pn$-QRPA model has been exploited using the Skyrme Hartree-Fock (SHF) plus BCS scheme developed by P. Sarrgiuren and co-workers Sarri2001 ; Sarri2001-b ; Sarri2005 ; Sarri2014 . They investigated the GT strength distribution in Zr and Mo isotopes Sarri2014 , the neutron-deficient Pb isotopes Sarri2005 , and the proton-rich Kr isotopes Sarri2001-b . For the neutron-rich nuclei, K. Yoshida also employed a model with Skyrme-type energy density functional (EDF) in the DQRPA Yoshi2008 ; Yoshi2013 .

The importance of the residual interaction was discussed in our early reports through calculations of both single- and double-$\beta$ decay transitions Ch93 ; pan . It should be noticed that these calculations were performed using spherical QRPA, which did not include the deformation explicitly. Primary aim of the present work is to perform the deformed QRPA (DQRPA) calculations with a realistic residual interaction, determined from the Brückner $G$-matrix with the CD Bonn potential. We utilized as a starting point the Skyrme-type mean field (MF) Stoitsov instead of the Woods-Saxon potential employed in the previous calculations Ha2015a ; Ha2015b ; Ha2016 . Therefore, this work is an extension of our recent works Ha2015a ; Ha2015b for the DQRPA, in which all effects of the deformation are consistently treated in the QRPA framework based on the Woods-Saxon mean field Ha2015a .

As an application of the present DQRPA model, we choose the GT transitions of two Mg isotopes, ²⁴Mg and ²⁶Mg, because they are known as well-deformed nuclei. Moreover, there are precise GT experimental data by ³He and $t$ beams in these nuclei Zegers08 ; Madey87 ; Zegers06 . We perform also the study of an almost spherical nucleus ¹⁸O in DHF calculation as a counter example to strongly deformed nuclei. Our paper is organized as follows. In Sec. II, we briefly explain the formalism including the deformation. The applications to the GT transition strength distributions for Mg isotopes are performed in Sec. III with detailed discussions on the choice of necessary parameters for $p-p$ and $p-h$ residual interactions in the present formalism. Conclusions are drawn with a short perspective for future works in Sec. IV.

Our calculation is carried out by using the QRPA Ha2015a , which adopts the SHF mean field Stoitsov for the mean field, and the residual interactions calculated by the Brückner $G$-matrix based on the CD Bonn potential. On top of the mean field, the pairing correlations are taken into account in the BCS approximation. We refer to our previous papers for the detailed calculations of the BCS wave functions including the tensor force (TF) Ha18-1 ; Ha18-2 . Hereafter, we briefly summarize the DQRPA model, which will be applied for calculations of the GT strength distributions.

In the following, we adopt the standard QRPA formalism based on the equation of motion for the following phonon operator, acting on the BCS ground state Ha2015a :

$${\cal Q}^{\dagger}_{m,K}=\sum_{\rho_{\alpha}\alpha\alpha^{\prime\prime}\rho_{\beta}\beta\beta^{\prime\prime}}[X^{m}_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}A^{\dagger}(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}K)-Y^{m}_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}{\tilde{A}}(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}K)]~{}.$$

(1)

where $\rho_{\alpha(\beta)}(=\pm 1)$ denotes the sign of the total angular momentum projection of the $\alpha$ state for the reflection symmetry. Here, we have introduced pair creation and annihilation operators, composed by two quasiparticles and defined as

$$A^{\dagger}(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}K)={[a^{\dagger}_{\alpha\alpha^{\prime\prime}}a^{\dagger}_{\beta\beta^{\prime\prime}}]}^{K},~{}~{}~{}{\tilde{A}}(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}K)={[a_{\beta\beta^{\prime\prime}}a_{\alpha\alpha^{\prime\prime}}]}^{K},$$

(2)

where $K$ is the quantum number associated with the projection of the intrinsic angular momentum on the symmetry axis, which is a good quantum number in the axially deformed nuclei. We note that parity is also treated as a good quantum number in the present approach. Here, $\alpha$ indicates a set of quantum numbers to specify the single-particle-state (SPS). Isospin of the real particle is denoted by the Greek letter with prime $(\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime})$ (see Eqs. (II) and (II)). Since our formalism is constructed to include the $np$ pairing correlations composed of $T=0$ and $T=1$ contributions, we have two different types of quasiparticles, quasi-proton and quasi-neutron, and the isospin of the quasiparticles cannot be clearly defined Ha2015a . The quantum number corresponding to the quasi-isospin is denoted by $(\alpha^{\prime\prime},\beta^{\prime\prime},\gamma^{\prime\prime},\delta^{\prime\prime})$, that is, with double-primes: in other words, these indices are $\alpha^{\prime\prime}(\beta^{\prime\prime},\gamma^{\prime\prime},\delta^{\prime\prime})=1,2$ and are used hereafter, instead of $\tau_{z}=\pm 1$ for neutrons and protons. Detailed discussion regarding the $T=0$ part has been done in our previous papers Ha18-1 ; Ha18-2 .

Within the quasi-boson approximation for the phonon operator, we obtain the following QRPA equation for describing the correlated DQRPA ground state:

	$\displaystyle\left(\begin{array}[]{cccccccc}A_{\alpha\beta\gamma\delta(K)}^{1111}&A_{\alpha\beta\gamma\delta(K)}^{1122}&A_{\alpha\beta\gamma\delta(K)}^{1112}&A_{\alpha\beta\gamma\delta(K)}^{1121}&B_{\alpha\beta\gamma\delta(K)}^{1111}&B_{\alpha\beta\gamma\delta(K)}^{1122}&B_{\alpha\beta\gamma\delta(K)}^{1112}&B_{\alpha\beta\gamma\delta(K)}^{1121}\\ A_{\alpha\beta\gamma\delta(K)}^{2211}&A_{\alpha\beta\gamma\delta(K)}^{2222}&A_{\alpha\beta\gamma\delta(K)}^{2212}&A_{\alpha\beta\gamma\delta(K)}^{2221}&B_{\alpha\beta\gamma\delta(K)}^{2211}&B_{\alpha\beta\gamma\delta(K)}^{2222}&B_{\alpha\beta\gamma\delta(K)}^{2212}&B_{\alpha\beta\gamma\delta(K)}^{2221}\\ A_{\alpha\beta\gamma\delta(K)}^{1211}&A_{\alpha\beta\gamma\delta(K)}^{1222}&A_{\alpha\beta\gamma\delta(K)}^{1212}&A_{\alpha\beta\gamma\delta(K)}^{1221}&B_{\alpha\beta\gamma\delta(K)}^{1211}&B_{\alpha\beta\gamma\delta(K)}^{1222}&B_{\alpha\beta\gamma\delta(K)}^{1212}&B_{\alpha\beta\gamma\delta(K)}^{1221}\\ A_{\alpha\beta\gamma\delta(K)}^{2111}&A_{\alpha\beta\gamma\delta(K)}^{2122}&A_{\alpha\beta\gamma\delta(K)}^{2112}&A_{\alpha\beta\gamma\delta(K)}^{2121}&B_{\alpha\beta\gamma\delta(K)}^{2111}&B_{\alpha\beta\gamma\delta(K)}^{2122}&B_{\alpha\beta\gamma\delta(K)}^{2112}&B_{\alpha\beta\gamma\delta(K)}^{2121}\\ &&&&&&&\\ -B_{\alpha\beta\gamma\delta(K)}^{1111}&-B_{\alpha\beta\gamma\delta(K)}^{1122}&-B_{\alpha\beta\gamma\delta(K)}^{1112}&-B_{\alpha\beta\gamma\delta(K)}^{1121}&-A_{\alpha\beta\gamma\delta(K)}^{1111}&-A_{\alpha\beta\gamma\delta(K)}^{1122}&-A_{\alpha\beta\gamma\delta(K)}^{1112}&-A_{\alpha\beta\gamma\delta(K)}^{1121}\\ -B_{\alpha\beta\gamma\delta(K)}^{2211}&-B_{\alpha\beta\gamma\delta(K)}^{2222}&-B_{\alpha\beta\gamma\delta(K)}^{2212}&-B_{\alpha\beta\gamma\delta(K)}^{2221}&-A_{\alpha\beta\gamma\delta(K)}^{2211}&-A_{\alpha\beta\gamma\delta(K)}^{2222}&-A_{\alpha\beta\gamma\delta(K)}^{2212}&-A_{\alpha\beta\gamma\delta(K)}^{2221}\\ -B_{\alpha\beta\gamma\delta(K)}^{1211}&-B_{\alpha\beta\gamma\delta(K)}^{1222}&-B_{\alpha\beta\gamma\delta(K)}^{1212}&-B_{\alpha\beta\gamma\delta(K)}^{1221}&-A_{\alpha\beta\gamma\delta(K)}^{1211}&-A_{\alpha\beta\gamma\delta(K)}^{1222}&-A_{\alpha\beta\gamma\delta(K)}^{1212}&-A_{\alpha\beta\gamma\delta(K)}^{1221}\\ -B_{\alpha\beta\gamma\delta(K)}^{2111}&-B_{\alpha\beta\gamma\delta(K)}^{2122}&-B_{\alpha\beta\gamma\delta(K)}^{2112}&-B_{\alpha\beta\gamma\delta(K)}^{2121}&-A_{\alpha\beta\gamma\delta(K)}^{2111}&-A_{\alpha\beta\gamma\delta(K)}^{2122}&-A_{\alpha\beta\gamma\delta(K)}^{2112}&-A_{\alpha\beta\gamma\delta(K)}^{2121}\\ \end{array}\right)$		(11)
	$\displaystyle\times\left(\begin{array}[]{c}{\tilde{X}}_{(\gamma 1\delta 1)K}^{m}\\ {\tilde{X}}_{(\gamma 2\delta 2)K}^{m}\\ {\tilde{X}}_{(\gamma 1\delta 2)K}^{m}\\ {\tilde{X}}_{(\gamma 2\delta 1)K}^{m}\\ \\ {\tilde{Y}}_{(\gamma 1\delta 1)K}^{m}\\ {\tilde{Y}}_{(\gamma 2\delta 2)K}^{m}\\ {\tilde{Y}}_{(\gamma 1\delta 2)K}^{m}\\ {\tilde{Y}}_{(\gamma 2\delta 1)K}^{m}\end{array}\right)=\hbar{\Omega}_{K}^{m}\left(\begin{array}[]{c}{\tilde{X}}_{(\alpha 1\beta 1)K}^{m}\\ {\tilde{X}}_{(\alpha 2\beta 2)K}^{m}\\ {\tilde{X}}_{(\alpha 1\beta 2)K}^{m}\\ {\tilde{X}}_{(\alpha 2\beta 1)K}^{m}\\ \\ {\tilde{Y}}_{(\alpha 1\beta 1)K}^{m}\\ {\tilde{Y}}_{(\alpha 2\beta 2)K}^{m}\\ {\tilde{Y}}_{(\alpha 1\beta 2)K}^{m}\\ {\tilde{Y}}_{(\alpha 2\beta 1)K}^{m}\end{array}\right)~{},$		(30)

where the amplitudes ${\tilde{X}}^{m}_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}$ and ${\tilde{Y}}^{m}_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}$ in Eq. (11) stand for forward and backward going amplitudes from the state ${\alpha\alpha^{\prime\prime}}$ to the state ${\beta\beta^{\prime\prime}}$ Ha2015a ,

The $A$ and $B$ matrices in Eq. (11) are given by

	$\displaystyle A_{\alpha\beta\gamma\delta(K)}^{\alpha^{\prime\prime}\beta^{\prime\prime}\gamma^{\prime\prime}\delta^{\prime\prime}}=$		$\displaystyle(E_{\alpha\alpha^{\prime\prime}}+E_{\beta\beta^{\prime\prime}})\delta_{\alpha\gamma}\delta_{\alpha^{\prime\prime}\gamma^{\prime\prime}}\delta_{\beta\delta}\delta_{\beta^{\prime\prime}\delta^{\prime\prime}}-\sigma_{\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}}\sigma_{\gamma\gamma^{\prime\prime}\delta\delta^{\prime\prime}}$
		$\displaystyle\times$	$\displaystyle\sum_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}[-g_{pp}(u_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}u_{\beta\beta^{\prime\prime}\beta^{\prime}}u_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}u_{\delta\delta^{\prime\prime}\delta^{\prime}}+v_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}v_{\beta\beta^{\prime\prime}\beta^{\prime}}v_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}v_{\delta\delta^{\prime\prime}\delta^{\prime}})~{}V_{\alpha\alpha^{\prime}\beta\beta^{\prime},~{}\gamma\gamma^{\prime}\delta\delta^{\prime}}$
		$\displaystyle-$	$\displaystyle g_{ph}(u_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}v_{\beta\beta^{\prime\prime}\beta^{\prime}}u_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}v_{\delta\delta^{\prime\prime}\delta^{\prime}}+v_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}u_{\beta\beta^{\prime\prime}\beta^{\prime}}v_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}u_{\delta\delta^{\prime\prime}\delta^{\prime}})~{}V_{\alpha\alpha^{\prime}\delta\delta^{\prime},~{}\gamma\gamma^{\prime}\beta\beta^{\prime}}$
		$\displaystyle-$	$\displaystyle g_{ph}(u_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}v_{\beta\beta^{\prime\prime}\beta^{\prime}}v_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}u_{\delta\delta^{\prime\prime}\delta^{\prime}}+v_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}u_{\beta\beta^{\prime\prime}\beta^{\prime}}u_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}v_{\delta\delta^{\prime\prime}\delta^{\prime}})~{}V_{\alpha\alpha^{\prime}\gamma\gamma^{\prime},~{}\delta\delta^{\prime}\beta\beta^{\prime}}],$

	$\displaystyle B_{\alpha\beta\gamma\delta(K)}^{\alpha^{\prime\prime}\beta^{\prime\prime}\gamma^{\prime\prime}\delta^{\prime\prime}}=$	$\displaystyle-$	$\displaystyle\sigma_{\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}}\sigma_{\gamma\gamma^{\prime\prime}\delta\delta^{\prime\prime}}$
		$\displaystyle\times$	$\displaystyle\sum_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}[g_{pp}(u_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}u_{\beta\beta^{\prime\prime}\beta^{\prime}}v_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}v_{\delta\delta^{\prime\prime}\delta^{\prime}}+v_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}v_{{\bar{\beta}}\beta^{\prime\prime}\beta^{\prime}}u_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}u_{{\bar{\delta}}\delta^{\prime\prime}\delta^{\prime}})~{}V_{\alpha\alpha^{\prime}\beta\beta^{\prime},~{}\gamma\gamma^{\prime}\delta\delta^{\prime}}$
		$\displaystyle-$	$\displaystyle g_{ph}(u_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}v_{\beta\beta^{\prime\prime}\beta^{\prime}}v_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}u_{\delta\delta^{\prime\prime}\delta^{\prime}}+v_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}u_{\beta\beta^{\prime\prime}\beta^{\prime}}u_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}v_{\delta\delta^{\prime\prime}\delta^{\prime}})~{}V_{\alpha\alpha^{\prime}\delta\delta^{\prime},~{}\gamma\gamma^{\prime}\beta\beta^{\prime}}$
		$\displaystyle-$	$\displaystyle g_{ph}(u_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}v_{\beta\beta^{\prime\prime}\beta^{\prime}}u_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}v_{\delta\delta^{\prime\prime}\delta^{\prime}}+v_{\alpha\alpha^{\prime\prime}\alpha^{\prime}}u_{\beta\beta^{\prime\prime}\beta^{\prime}}v_{\gamma\gamma^{\prime\prime}\gamma^{\prime}}u_{\delta\delta^{\prime\prime}\delta^{\prime}})~{}V_{\alpha\alpha^{\prime}\gamma\gamma^{\prime},~{}\delta\delta^{\prime}\beta\beta^{\prime}}],$

where $\sigma_{\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}}$ = 1 if $\alpha=\beta$ and $\alpha^{\prime\prime}$ = $\beta^{\prime\prime}$, otherwise $\sigma_{\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}}$ = $\sqrt{2}$ Ch93 . The $u$ and $v$ coefficients are determined from the gap equations. The $g_{pp}$ and $g_{ph}$ stand for particle-particle and particle-hole renormalization factors for the residual interactions in Eqs. (II) and (II). The two-body interactions $V_{\alpha\beta,~{}\gamma\delta}$ and $V_{\alpha\delta,~{}\gamma\beta}$ are particle-particle and particle-hole matrix elements of the residual $N$-$N$ interaction $V$, respectively, which are calculated from the $G$-matrix as solutions of the Bethe-Goldstone equation from the CD Bonn potential.

The two-body interactions $V_{\alpha\beta,~{}\gamma\delta}$ and $V_{\alpha\delta,~{}\gamma\beta}$ are, respectively, $p-p$ and $p-h$ matrix elements of the residual $N-N$ interaction in the deformed state. They are calculated from the $G$-matrix in the spherical basis as follows

	$\displaystyle V_{\alpha\alpha^{\prime}\beta\beta^{\prime},~{}\gamma\gamma^{\prime}\delta\delta^{\prime}}=-$		$\displaystyle\sum_{J}\sum_{abcd}F_{\alpha a{\bar{\beta}}b}^{JK}F_{\gamma c{\bar{\delta}}d}^{JK}G(a\alpha^{\prime}b\beta^{\prime}c\gamma^{\prime}d\delta^{\prime},J)~{},$		(33)
	$\displaystyle V_{\alpha\alpha^{\prime}\delta\delta^{\prime},\gamma\gamma^{\prime}\beta\beta^{\prime}}=$		$\displaystyle\sum_{J}\sum_{abcd}F_{\alpha a{\delta}d}^{JK}F_{\gamma c{\beta}b}^{JK}G(a\alpha^{\prime}d\delta^{\prime}c\gamma^{\prime}b\beta^{\prime},J)~{},$
	$\displaystyle V_{\alpha\alpha^{\prime}\gamma\gamma^{\prime},\delta\delta^{\prime}\beta\beta^{\prime}}=$		$\displaystyle\sum_{J}\sum_{abcd}F_{\alpha a{\gamma}c}^{JK}F_{\beta b{\delta}d}^{JK}G(a\alpha^{\prime}c\gamma^{\prime}d\delta^{\prime}b\beta^{\prime},J)~{}.$

Here $a$ and $\alpha$ indicates spherical and deformed SPS, respectively. We note that the quasi-isospin $\alpha^{\prime},\beta^{\prime},\gamma^{\prime}$ and $\delta^{\prime}$ in the $G$-matrix is replaced by total isospin ($T=0$ or $T=1$) of the two-body interaction in the isospin representation. We use $F_{\alpha a{\bar{\beta}}b}^{JK}=B_{a}^{\alpha}B_{b}^{\beta}C_{j_{a}\Omega_{\alpha}j_{b}\Omega_{\beta}}$ for $K=\Omega_{\alpha}+\Omega_{\beta}$. The expansion coefficient $B_{\alpha}^{a}$ is defined as

$$|\alpha\Omega_{\alpha}>=\sum_{a}B_{a}^{\alpha}|a\Omega_{\alpha}>~{},~{}B_{a}^{\alpha}=\sum_{Nn_{z}}C_{l\Lambda{1\over 2}\Sigma}^{j\Omega_{\alpha}}A_{Nn_{z}\Lambda}^{N_{0}l}b_{Nn_{z}\Sigma},$$

(34)

with the Clebsch-Gordan coefficient $C_{l\Lambda{1\over 2}\Sigma}^{j\Omega_{\alpha}}$, the spatial overlap integral $A_{Nn_{z}\Lambda}^{N_{0}l}$, and the eigenvalues obtained from the total Hamiltonian in the deformed basis $b_{Nn_{z}\Sigma}$. Detailed formulas regarding the transformation of Eq. (7) can be found in Ref. Ha2015a .

Our DQRPA equation has a very general form because we include the deformation as well as two kinds of pairing correlations, $(T=1,S=0)$ and $(T=0,S=1)$, in the model. If we switch off the $np$ pairing, all off-diagonal terms in the $A$ and $B$ matrices in Eq. (11) disappear with the replacement of 1 and 2 into $p$ and $n$. Then the DQRPA equation is decoupled into $pp+nn+pn+np$ DQRPA equations. $pp+nn$ QRPA can describe charge-conserving reactions such as the electromagnetic (EM) transitions, including magnetic dipole (M1) transition, between the initial and final states in the same nucleus, while $np+pn$ QRPA describes charge-exchange reactions like the GT${(+/-)}$ transitions between mother and daughter nuclei having different proton and neutron numbers, $(N,Z)\rightarrow(N\pm 1,Z\mp 1)$. If we assume spherical symmetry, our equations are reduced to the QRPA equations in Ref. Ch93 . If we neglect the $np$ pairing for the nuclei considered in this work, i.e. take only 1212 terms, Eq. (11) becomes the proton-neutron DQRPA as in Ref. saleh .

The GT transition operator ${\hat{\textrm{O}}}_{1\mu}^{\pm}$ is defined by

$${\hat{\textrm{O}}}_{1\mu}^{-}=\sum_{\alpha\beta}<\alpha p|\tau^{+}\sigma_{K}|\beta n>c_{\alpha p}^{\dagger}{\tilde{c}}_{\beta n},~{}{\hat{\textrm{O}}}_{1\mu}^{+}={({\hat{T}}_{1\mu}^{-})}^{\dagger}={(-)}^{\mu}{\hat{\textrm{O}}}_{1,-\mu}^{-}.$$

(35)

Detailed calculation for the transformation from the intrinsic frame to the nuclear laboratory system were presented in Ref. Ha2015a . The ${\hat{\textrm{O}}}^{\pm}$ transition amplitudes from the ground state of an initial (parent) nucleus to the excited state of a final (daughter) nucleus, i.e. the one phonon state $|K^{+},m\rangle$, are written as

	$\displaystyle<K^{+},m|{\hat{\textrm{O}}}_{K}^{-}|~{}QRPA>$		(36)
	$\displaystyle=\sum_{\alpha\alpha^{\prime\prime}\rho_{\alpha}\beta\beta^{\prime\prime}\rho_{\beta}}{\cal N}_{\alpha\alpha^{\prime\prime}\rho_{\alpha}\beta\beta^{\prime\prime}\rho_{\beta}}<\alpha\alpha^{\prime\prime}p\rho_{\alpha}|\sigma_{K}|\beta\beta^{\prime\prime}n\rho_{\beta}>[u_{\alpha\alpha^{\prime\prime}p}v_{\beta\beta^{\prime\prime}n}X_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}^{m}+v_{\alpha\alpha^{\prime\prime}p}u_{\beta\beta^{\prime\prime}n}Y_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}^{m}],$
	$\displaystyle<K^{+},m|{\hat{\textrm{O}}}_{K}^{+}|~{}QRPA>$
	$\displaystyle=\sum_{\alpha\alpha^{\prime\prime}\rho_{\alpha}\beta\beta^{\prime\prime}\rho_{\beta}}{\cal N}_{\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}}<\alpha\alpha^{\prime\prime}p\rho_{\alpha}|\sigma_{K}|\beta\beta^{\prime\prime}n\rho_{\beta}>[u_{\alpha\alpha^{\prime\prime}p}v_{\beta\beta^{\prime\prime}n}Y_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}^{m}+v_{\alpha\alpha^{\prime\prime}p}u_{\beta\beta^{\prime\prime}n}X_{(\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime})K}^{m}]~{},$

where $|~{}QRPA>$ denotes the correlated QRPA ground state in the intrinsic frame and the nomalization factor is given as ${\cal N}_{\alpha\alpha^{\prime\prime}\beta\beta^{\prime\prime}}(J)=\sqrt{1-\delta_{\alpha\beta}\delta_{\alpha^{\prime\prime}\beta^{\prime\prime}}(-1)^{J+T}}/(1+\delta_{\alpha\beta}\delta_{\alpha^{\prime\prime}\beta^{\prime\prime}}).$

Here, we use the following pairing interaction,

$\displaystyle{\cal H}_{pair}({\bf r})={1\over 2}V_{0}[1-V_{1}({\rho/\rho_{0}})^{\gamma}],$

(37)

where the parameter $V_{0}$ for ¹⁸O is adjusted for fixing the GT peak position, and the other $V_{0}$, $V_{1}$ and $\gamma$ are taken from Ref. Stoitsov . The results of the parameters and the pairing gaps obtained with the pairing window associated with an energy cut-off of 60 MeV, are summarized in Table 1 for the Skyrme EDF parameter sets adopted in the present work.

This study is based on the mean field obtained with the axially deformed Skyrme Hartree-Fock-Bogoliubov (SHFB) approach using a harmonic oscillator basis Stoitsov . The particle model space, for the nuclei considered here, includes states up to $N=5\hbar\omega$ and $N=10\hbar\omega$, respectively, for the deformed and spherical basis. In the DQRPA, we adopt the wave functions and the SP energies from the SHFB equations. We utilize the SP energies from the deformed SHF mean field, and the $G$-matrix for the two-body interaction in Eq. (II) and (II) is also calculated by using the wave functions from the deformed SHF.

In Fig. 1, we illustrate the SPS evolution of protons and neutrons with the deformation. We note that the SP energy splitting of either the $d_{3/2}$ state or the $d_{5/2}$ state, induced by deformation in ^24,26Mg, smears out the shell closures, and the magic numbers $N=8$ and 20 are disappeared. Instead, in the current work, $N=12$ appears as a proton quasi-magic number for ${}^{24}_{12}$Mg₁₂ and ${}^{26}_{12}$Mg₁₄ because of the relatively large energy gaps above the $N=12$ shell induced by the strong deformations as tabulated in Table 2. We note that the pairing interaction in Eq. (10) may smear the occupation probabilities, but the smearing is not significant, as shown in Fig. 4.

Another interesting point is that the SP energies of $\nu d_{5/2}$ and $\nu d_{3/2}$ orbitals are slightly higher in energy as the neutron number is increased while those of protons $\pi d_{5/2}$ and $\pi d_{3/2}$ orbitals are decreased. As a consequence, the Fermi energy gap of protons and neutrons for ²⁶Mg and ¹⁶O is about 6 MeV, but for ²⁴Mg it is about 1.5 MeV. Interestingly, in ^24,26Mg, the three states, $\nu({5/2}^{+}_{1})$ and $\nu({1/2}^{+}_{3})$ as well as $\nu({1/2}^{-}_{3})$, are almost degenerate.

We investigated the GT strength distributions of largely deformed s-d shell nuclei, ²⁴Mg, ²⁶Mg, and of an almost spherical nucleus, ¹⁸O, in the framework of Deformed QRPA (DQRPA). The present calculation adopts the SHF mean field with the residual interaction derived the $G$-matrix calculation. We found that major part of the experimental data are better explained than in the previous calculations based on the Woods-Saxon potential Ha2016 . We found that ²⁴Mg and ²⁶Mg are strongly deformed, and a sub-magic shell at $N=12$ is found and plays an important role to describe the GT strength distributions of these nuclei. The cumulated sums of GT_± strengths of ²⁴Mg, ²⁶Mg are studied in the low-energy region where the experimental data are available. Our SHFB+QRPA results give a good account of the sum rule values with a quenching factor $q=0.684$ for the GT operator.

We also confirmed similarities of the GT strength distributions of $N=Z$ nuclei to those of $N=Z+2$ nuclei. It implies that the low-lying GT states in $N=Z$ $(N=Z+2)$ nuclei are enhanced mainly by the $p-h$ ($p-p$) interaction, and relatively small GT strengths appear in the higher energy region in the case of the nuclei studied in this article. We argued that the strong deformation makes it so that the relevant residual interaction is a combination of $p-p$ and $p-h$ interactions, as shown in ²⁴Mg and ²⁶Mg, due to the smearing of Fermi surface by the pairing correlations. On the other hand, this phenomenon is not significant in the case of ¹⁸O nucleus, since it is almost spherical.