Isospin symmetry of $T_z=\pm 3∕2\to \pm 1∕2$ Gamow-Teller transitions in $A=41$ nuclei

Under the assumption that isospin $T$ is a good quantum number, isobaric analog states and various analogous transitions are expected in isobars with mass number $A$. The strengths of $T_z=\pm 3∕2\to \pm 1∕2$ analogous Gamow-Teller (GT) transitions and analogous $M1$ transitions within the $A=41$ isobar quartet are compared in detail. The $T_z=+3∕2\to +1∕2$ GT transitions from the $J^{\pi }={3∕2}^{+}$ ground state of ${}^{41}\mathrm{K}$ leading to excited $J^{\pi }={1∕2}^{+}$, ${3∕2}^{+}$, and ${5∕2}^{+}$ states in ${}^{41}\mathrm{Ca}$ were measured using the $({}^3\mathrm{He},t)$ charge-exchange reaction. With a high energy resolution of $35\mathrm{keV}$, many fragmented states were observed, and the GT strength distribution was determined up to $10\mathrm{MeV}$ excitation energy $\left(E_x\right)$. The main part of the strength was concentrated in the $E_x=4–6\mathrm{MeV}$ region. A shell-model calculation could reproduce the concentration, but not so well details of the strength distribution. The obtained distribution was further compared with two results of ${}^{41}\mathrm{Ti}$ $\beta$ decay studying the analogous $T_z=-3∕2\to -1∕2$ GT strengths. They reported contradicting distributions. One-to-one correspondences of analogous transitions and analog states were assigned up to $E_x=6\mathrm{MeV}$ in the comparison with one of these ${}^{41}\mathrm{Ti}$ $\beta$-decay results. Combining the spectroscopic information of the analog states in ${}^{41}\mathrm{Ca}$ and ${}^{41}\mathrm{Sc}$, the most probable $J^{\pi }$ values were deduced for each pair of analog states. It was found that ${5∕2}^{+}$ states carry the main part of the observed GT strength, while much less GT strength was carried by ${1∕2}^{+}$ and ${3∕2}^{+}$ states. The gross features of the GT strength distributions for each $J$ were similar for the isospin analogous $T_z=\pm 3∕2\to \pm 1∕2$ transitions, but the details were somewhat different. From the difference of the distributions, isospin-asymmetry matrix elements of $\approx 8\mathrm{keV}$ were deduced. The Coulomb displacement energy, which is sensitive to the configuration of states, showed a sudden increase of about $50\mathrm{keV}$ at the excitation energy of $3.8\mathrm{MeV}$. The strengths of several $M1$ transitions to the IAS in ${}^{41}\mathrm{Ca}$ were compared with the strengths of analogous GT transitions. It was found that ratios of the $M1$ and GT transition strengths were similar, suggesting that the contributions of the $\ell \tau$ term in $M1$ transitions are small.

Figure 1

(Color online) Schematic view of the isopin analog states and analogous transitions in the $A=41$, $T_z=\pm 3∕2$ and $\pm 1∕2$ isobar system. The Coulomb displacement energies are removed so that the isospin symmetry of the states and transitions become clearer. The type of the reaction or decay is shown along the arrow indicating the transition.

Figure 2

(Color online) The ${}^{41}\mathrm{K}({}^3\mathrm{He},t){}^{41}\mathrm{Ca}$ spectrum at 0° with an angular range up to 0.5°. A high resolution of $35\mathrm{keV}$ has been achieved. The major GT states are indicated by their excitation energies. For other details, see text.

Figure 3

(Color online) Experimental and shell-model $B\left(\mathrm{GT}\right)$ distributions. (a) $B\left(\mathrm{GT}\right)$ distribution from the present ${}^{41}\mathrm{K}({}^3\mathrm{He},t){}^{41}\mathrm{Ca}$ measurement. (b) $B\left(\mathrm{GT}\right)$ distribution from a shell-model (SM) calculation for all allowed $J$. $B\left(\mathrm{GT}\right)$ distribution from a SM calculation (c) for $J=1∕2$, (d) for $J=3∕2$, and (e) for $J=5∕2$. Note the change in size of the panels.

Figure 4

(Color online) Experimental $B\left(\mathrm{GT}\right)$ distributions from $T_z=\pm 3∕2\to \pm 1∕2$ isospin mirror transitions. (a) $B\left(\mathrm{GT}\right)$ distribution from the present ${}^{41}\mathrm{K}({}^3\mathrm{He},t){}^{41}\mathrm{Ca}$ reaction. (b) $B\left(\mathrm{GT}\right)$ distribution from the ${}^{41}\mathrm{Ti}\to {}^{41}\mathrm{Sc}$ $\beta$ decay reported in Ref. 18. (c) $B\left(\mathrm{GT}\right)$ distribution from the ${}^{41}\mathrm{Ti}\to {}^{41}\mathrm{Sc}$ $\beta$ decay reported in Ref. 19.

Figure 5

A correlation of $B\left(\mathrm{GT}\right)$ values between analogous $T_z\pm 3∕2\to \pm 1∕2$ transitions. The $-3∕2\to -1∕2$ $B{\left(\mathrm{GT}\right)}_{\beta }$ values are from ${}^{41}\mathrm{Ti}$ $\beta$ decay [18] and the $+3∕2\to +1∕2$ $B{\left(\mathrm{GT}\right)}_{\mathrm{het}}$ values are from the present ${}^{41}\mathrm{K}({}^3\mathrm{He},t)$ measurement. Analog states with ideal correlation would follow the dashed line.

Figure 6

(Color online) The $T_z=+3∕2\to +1∕2$ $B\left(\mathrm{GT}\right)$ distributions deduced from the ${}^{41}\mathrm{K}({}^3\mathrm{He},t)$ measurement for the ${}^{41}\mathrm{Ca}$ states with (a) $J^{\pi }={1∕2}^{+}$, (b) $J^{\pi }={3∕2}^{+}$, and (c) $J^{\pi }={5∕2}^{+}$.

Figure 7

(Color online) The $T_z=-3∕2\to -1∕2$ $B\left(\mathrm{GT}\right)$ distributions deduced from the ${}^{41}\mathrm{Ti}$ $\beta$-decay measurement for the ${}^{41}\mathrm{Sc}$ states with (a) $J^{\pi }={1∕2}^{+}$, (b) $J^{\pi }={3∕2}^{+}$, and (c) $J^{\pi }={5∕2}^{+}$.

Figure 8

The difference of excitation energies $\Delta E_x$ for the corresponding states in ${}^{41}\mathrm{Sc}$ and ${}^{41}\mathrm{Ca}$. The $\Delta E_x$ values are plotted for pairs of states with (a) $J=1∕2$, (b) $3∕2$, and (c) $5∕2$. A clear sudden increase of $\Delta E_x$ values is observed at $E_x=3.8\mathrm{MeV}$ for the pairs of $J=1∕2$ and $5∕2$ states. The dotted lines are drawn to guide the eye.