Gamow-Teller transitions from the ${}^{14}\mathrm{N}$ ground state to the ${}^{14}\mathrm{C}$ ground and excited states

Gamow-Teller transitions from the ${}^{14}\mathrm{N}$ ground state to the ${}^{14}\mathrm{C}$ ground and excited states were investigated, based on the model of antisymmetrized molecular dynamics. The calculated strengths for the allowed transitions to the $0^{+}$, $1^{+}$, and $2^{+}$ states of ${}^{14}\mathrm{C}$ were compared with the experimental data measured by high-resolution charge-exchange reactions. The calculated GT transition to the $2_1^{+}$ state is strong while those to the $0_{2,3}^{+}$ and $2_{2,3}^{+}$ states having dominant $2\hslash \omega$ excited configurations are relatively weak. The present calculation cannot describe the anonymously long lifetime of ${}^{14}\mathrm{C}$, though the strength of the ${}^{14}\mathrm{C}$ ground state is somewhat suppressed because of the cluster (many-body) correlation in the ground states of ${}^{14}\mathrm{C}$ and ${}^{14}\mathrm{N}$.

Figure 1

The $B\left(\mathrm{GT}\right)$ distributions for transitions from the $1^{+}$ ground state of ${}^{14}\mathrm{N}$ to $J^{\pi }=0^{+}$,$1^{+}$, and $2^{+}$ states of ${}^{14}\mathrm{C}$. (a) The theoretical values calculated with the AMD+VAP using the set A interaction, (b) those calculated with the $\beta$-$\gamma$ constraint AMD+GCM, (c) the experimental data taken from Refs. [8, 26], and (d) the theoretical results from Ref. [9] of the NCSM calculation with the $6\hslash \omega$ model space using the effective interactions derived from Argonne V8' interaction.

Figure 2

Density distribution for the intrinsic wave functions ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{J_k^{\pi }}\right)$ of ${}^{14}\mathrm{C}$ and ${}^{14}\mathrm{N}$ obtained with the AMD+VAP using the set A interaction. $x$, $y$, and $z$ axes are chosen as $\langle x^2\rangle \le \langle y^2\rangle \le \langle z^2\rangle$ and the matter density integrated along the $x$ axis is plotted on the $z$-$y$ plane.

Figure 3

Occupation probability $\langle P_Q\rangle$ for protons, neutrons, and total nucleons calculated with the AMD+VAP using the set A interaction. The calculated $\langle P_Q\rangle$ values are plotted as functions of $\Delta Q\equiv Q-Q_{\min }$ measured from the lowest allowed HO quanta $Q_{\min }$.

Figure 4

Energy expectation values calculated with the $\beta$-$\gamma$ constraint AMD. (a) Energy for the positive-parity state projected from ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$ of ${}^{14}\mathrm{C}$ without the spin projection. (b) and (c) Energy for the $0^{+}$ and $2^{+}$ states of ${}^{14}\mathrm{C}$ projected from ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$. (d) Energy for the positive-parity state projected from ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$ of ${}^{14}\mathrm{N}$ without the spin projection. (e) Energy for the $1^{+}$ state of ${}^{14}\mathrm{N}$ projected from ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$.

Figure 5

Density distribution for the intrinsic wave functions ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$ of ${}^{14}\mathrm{C}$ obtained with the $\beta$-$\gamma$ constraint AMD using the set A interaction. The densities for the $J^{\pi }=0^{+}$ energy minimum $({\beta }_{\min }cos{\gamma }_{\min },{\beta }_{\min }sin{\gamma }_{\min })=(0.225,0.130)$ state and the spherical $\beta =0$, prolate $(\beta cos\gamma ,\beta sin\gamma )=(0.4,0.0)$, triaxial $(0.325,0.130)$, and oblate $(0.125,0.216)$ states are shown. $x$, $y$, and $z$ axes are chosen as $\langle x^2\rangle \le \langle y^2\rangle \le \langle z^2\rangle$ and the matter density integrated along the $x$ axis is plotted on the $z$-$y$ plane.

Figure 6

Density distribution for the intrinsic wave functions ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$ of ${}^{14}\mathrm{N}$ obtained with the $\beta$-$\gamma$ constraint AMD using the set A interaction. The densities for the $J^{\pi }=1^{+}$ energy minimum $({\beta }_{\min }cos{\gamma }_{\min },{\beta }_{\min }sin{\gamma }_{\min })=(0.250,0.087)$ state, and the spherical $\beta =0$, prolate $(\beta cos\gamma ,\beta sin\gamma )=(0.4,0.0)$, triaxial $(0.325,0.130)$, and oblate $(0.125,0.216)$ states are shown. $x$, $y$, and $z$ axes are chosen as $\langle x^2\rangle \le \langle y^2\rangle \le \langle z^2\rangle$, and the matter density integrated along the $x$ axis is plotted on the $z$-$y$ plane.

Figure 7

$B\left(\mathrm{GT}\right)$ values evaluated by the GT matrix element obtained using the single $\beta$-$\gamma$ constraint AMD wave function ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$ for ${}^{14}\mathrm{N}$ and that for ${}^{14}\mathrm{C}$. (a) and (b) $B\left(\mathrm{GT}\right)$ values for the $J^{\pi }=0^{+}$ and $2^{+}$ states of ${}^{14}\mathrm{C}$ projected from ${\Phi }_{\mathrm{AMD}}\left({\mathbf{Z}}^{(\beta ,\gamma )}\right)$. The initial state is the $1^{+}$ energy minimum state of ${}^{14}\mathrm{N}$ at $({\beta }_{\min }cos{\gamma }_{\min },{\beta }_{\min }sin{\gamma }_{\min })=(0.250,0.087)$. Panels (c) and (d) are the same as panels (a) and (b) but the initial ${}^{14}\mathrm{N}$ state is the $1^{+}$ state projected from the spherical $\beta =0$ wave function.