High-resolution study of $T_z=+2\to +1$ Gamow-Teller transitions in the ${}^{44}$Ca(${}^3$He,$t$)${}^{44}$Sc reaction

In order to study the Gamow-Teller (GT) transitions from the $T_z=+2$ nucleus ${}^{44}$Ca to the $T_z=+1$ nucleus ${}^{44}$Sc, where $T_z$ is the $z$ component of isospin $T$, we performed the ($p,n$)-type (${}^3\mathrm{He},t$) charge-exchange (CE) reaction at 140 MeV/nucleon and the scattering angles $0^{\circ }$ and $2.5^{\circ }$. An energy resolution of 28 keV, that was realized by applying matching techniques to the magnetic spectrometer system, allowed the study of fragmented states. The GT transition strengths, $B$(GT), were derived up to the excitation energy ($E_x$) of 13.7 MeV assuming the proportionality between cross sections and $B$(GT) values. The total sum of $B$(GT) values in discrete states was 3.7, which was 31$\%$ of the sum-rule-limit value of 12. Shell model calculations using the GXPF1J interaction could reproduce the gross features of the experimental $B$(GT) distribution, but not the fragmentation of the strength. By introducing the concepts of isospin, properties of isospin analogous transitions and states were investigated. (i) Assuming isospin symmetry, the $T_z=+2\to +1$ and $T_z=-2\to -1$ mirror GT transitions should have the same properties, where the latter can be studied in the $\beta$ decay of ${}^{44}$Cr to ${}^{44}$V. First, we confirmed that the $\beta$-decay half-life $T_{1/2}$ of ${}^{44}$Cr can be reproduced using the $B$(GT) distribution from the ${}^{44}$Ca(${}^3\mathrm{He},t$) measurement. Then, the $0^{\circ }$, (${}^3\mathrm{He},t$) spectrum was modified to deduce the “$\beta$-decay spectrum” and it was compared with the delayed-proton spectrum from the ${}^{44}$Cr $\beta$ decay. The two spectra were mostly in agreement for the GT excitations, but suppression of the proton decay was found for the $T=2$ isobaric analog state (IAS). (ii) Starting from the $T=2$ ground state of ${}^{44}$Ca, the (${}^3\mathrm{He},t$) can excite GT states (state populated by GT transitions) with $T=1$, 2, and 3. On the other hand, the ${}^{44}$Ca($p,p^{\prime }$) reaction can excite spin-$M1$ states (states populated by spin-$M1$ transitions) with $T=2$ and 3 that are analogous to the $T=2$ and 3 GT states, respectively. By comparing the spectra from these two reactions, a $T$ value of 2 is suggested for several GT states in the $E_x=11.5$–13.7 MeV region. (iii) It has been suggested that the $T=2$, $J^{\pi }=0^{+}$ double isobaric analog state (DIAS) at 9.338 MeV in the $T_z=0$ nucleus ${}^{44}$Ti forms an isospin-mixed doublet with a subsidiary $0^{+}$ state at 9.298 MeV. Since no corresponding state was found in the $T_z=+1$ nucleus ${}^{44}$Sc, we suggest $T=0$ for the subsidiary state.

Figure 1

Schematic view of the analog states (connected by dashed lines) and analogous transitions in the mass $A=44$, $T_z=+3$, $+$2, $+$1, 0, −1, and $-2$ isobaric system. The Coulomb displacement energies are removed so that the isospin symmetry of analog states and analogous transitions becomes clearer. In this scheme, the inclined arrows show the $0^{+}\to 1^{+}$, GT transitions caused by the $\sigma \tau$-type operator from the ground states of mirror nuclei ${}^{44}$Ca and ${}^{44}$Cr. On the other hand, the analogous $0^{+}\to 0^{+}$ Fermi transitions to the IASs governed by the $\tau$-type operator are shown by the horizontal arrows. The vertical arrows show the $0^{+}\to 1^{+}$ transitions caused by inelastic-type reactions such as ($p,p^{\prime }$) or ($e,e^{\prime }$) on ${}^{44}$Ca. The isobaric analog state in ${}^{44}$Ti is called the double isobaric analog state (DIAS).

Figure 2

The $0^{\circ }$, ${}^{44}$Ca(${}^3\mathrm{He},t$)${}^{44}$Sc spectrum on two scales. The events within the range of scattering angles $\Theta \le 0.5^{\circ }$ are included. (a) The full count range spectrum. Prominent peaks are observed mainly in the region below 6 MeV. Most of them are populated in $\Delta L=0$ transitions. (b) The vertical scale is magnified by one order of magnitude, and a fine structure of many states is observed up to $E_x=14$ MeV. Major states populated in $\Delta L=0$ transitions are indicated by their excitation energies in MeV.

Figure 3

A comparison of the experimental and theoretical $B$(GT) strength distributions with (a) the $B$(GT) strength distribution derived from the ${}^{44}$Ca(${}^3\mathrm{He},t$) measurement, and (b) the $B$(GT) strength distribution from the shell model (SM) calculation using the effective interaction GXPF1J. A quenching factor of ${\left(0.74\right)}^2$ is included.

Figure 4

A comparison of the cumulative sums (CSs) of $B$(GT) strengths from the experimental ${}^{44}$Ca(${}^3\mathrm{He},t$) measurement and the SM calculation using the effective interaction GXPF1J. A quenching factor of ${\left(0.74\right)}^2$ is included in the SM calculation.

Figure 5

(a) The low-excitation energy region of the ${}^{44}$Ca(${}^3\mathrm{He},t$)${}^{44}$Sc spectrum for events with scattering angles $\Theta \le 0.5^{\circ }$. Major states with $\Delta L=0$ character are indicated by their excitation energies in MeV. The spectrum has been modified to include the decrease of the cross section caused by the factor $F(q,\omega )$ in Eq. (2). Therefore, the heights of GT states are almost proportional to $B$(GT) values. (b) The $f$ factor for the ${}^{44}$Cr $\beta$ decay normalized to unity at $E_x=0$ MeV. (c) The modified ${}^{44}$Ca(${}^3\mathrm{He},t$)${}^{44}$Sc spectrum that is obtained by multiplying the (${}^3\mathrm{He},t$) spectrum given in panel (a) with the $f$ factor shown in panel (b). (d) The deduced $\beta$-decay spectrum of ${}^{44}$Cr after making the correction of the IAS strength. Note that the IAS is stronger by a factor of $R^2/{\lambda }^2$ in a $\beta$ decay compared to the corresponding CE reaction (see text).

Figure 6

(a) The delayed charged-particle spectrum from the ${}^{44}$Cr $\beta$ decay [21]. The low-energy bump indicated by “low-E BG” originates in the emission of ${\beta }^{+}$ particles, while other peaks are from the delayed-proton emission, where the pileup of $\beta$-particle energy broadens the peaks and makes the high-energy tails of the peaks wide. The peaks are identified by the energy given in Ref. [21]. The energy resolution ($\Delta E$) is about 150 keV (FWHM). (b) The $\beta$-decay spectrum deduced from the ${}^{44}$Ca(${}^3\mathrm{He},t$) spectrum. The spectrum shown in Fig. 5d was convoluted with the $\Delta E$ of 150 keV of the delayed-proton spectrum shown in panel (a). The energy scale is shifted by 1.8 MeV to get a good agreement with the delayed-proton spectrum. The vertical scale is normalized so that the 3.67 MeV peak consisting of a few GT states has approximately the same height as the corresponding 1.74 MeV peak in panel (a). The solid line was obtained by a simple convolution of the 150 keV (FWHM) width. The dashed, short-dashed, and dotted lines were obtained by reducing the IAS strength down to 40$\%$, 30$\%$, and 20$\%$, respectively, in order to simulate the suppression of the proton emission from the IAS.

Figure 7

(a) The ${}^{44}$Ca(${}^3\mathrm{He},t$) spectrum of the $E_x=10$–16 MeV region for events with scattering angles $\Theta \le 0.5^{\circ }$. States with or probably with $\Delta L=0$ are indicated by “0” and “(0),” respectively. (b) The ${}^{44}$Ca($p,p^{\prime }$) spectrum measured at IUCF [59]. The measurements were carried out at $E_p=200$ MeV and at very forward angles including $0^{\circ }$ using the K600 spectrometer in transmission mode. The excitations in the $E_x>8.3$ MeV region in ${}^{44}$Ca were studied. We see several pairs of sharp peaks at the corresponding energies in the spectra. Note that the ordinates of both figures begin with finite counts to illustrate the structured part clearly.

Figure 8

The $0^{\circ }$, ${}^{44}$Ca(${}^3\mathrm{He},t$)${}^{44}$Sc spectrum showing the region around the IAS at 2.779 MeV.