$\mathit{ft}$ value of the $0^{+}\to 0^{+}$ ${\beta }^{+}$ decay of ${}^{32}\mathrm{Ar}$: A measurement of isospin symmetry breaking in a superallowed decay

We determined the absolute branch of the $T=2$ superallowed decay of ${}^{32}\mathrm{Ar}$ by detecting the ${\beta }^{+}$-delayed protons and γ decays of the daughter state. We obtain $b_{\mathrm{SA}}^{\beta }=(22.71\pm 0.16)\%$, which represents the first determination of a proton branch to better than 1%. Using this branch along with the previously determined ${}^{32}\mathrm{Ar}$ half-life and energy release, we determined $\mathit{ft}=(1552\pm 12)$ s for the superallowed decay. This $\mathit{ft}$ value, together with the corrected $\mathcal{F}t$ value extracted from previously known $T=1$ superallowed decays, yields a measurement of the isospin symmetry breaking correction in ${}^{32}\mathrm{Ar}$ decay ${\delta }_C^{\exp }=(2.1\pm 0.8)\%$. This can be compared to a theoretical calculation ${\delta }_C=(2.0\pm 0.4)\%$. As by-products of this work, we determined the γ and proton branches for the decay of the lowest $T=2$ state of ${}^{32}\mathrm{Cl}$, made a precise determination of the total proton branch and relative intensities of proton groups that leave ${}^{31}\mathrm{S}$ in its first excited state and deduced an improved value for the ${}^{32}\mathrm{Cl}$ mass.

Figure 1

Predicted isospin-symmetry-breaking correction, ${\delta }_C$, versus $Z$ for a number of superallowed decays (from Ref. [6]) as well as our calculation for ${}^{32}\mathrm{Ar}$.

Figure 2

Schematic view of the experiment (to scale) with the Ge and Si detectors labeled. The ${}^{32}\mathrm{Ar}$ ion beam first passes through the ${\mathrm{D}}_1$ PIN diode (off scale to the right), then goes through the ${\mathrm{D}}_2$ detector before stopping in the middle of ${\mathrm{D}}_3$. Half of clovers 2 and 3 have been cut away to show the position of the particle detectors in the beam line; this cutaway view is magnified in the top-right balloon to clearly show the relative Si detector positions.

Figure 3

$E_1$ versus TOF${}_1$ spectrum from a single run. (Top) Surface plot where the ${}^{32}\mathrm{Ar}$ group has been truncated to show the other groups more clearly. (Bottom) Vertical projection showing Region 3.1, which contains the main ${}^{32}\mathrm{Ar}$ group, and Region 3.2 that surrounds Region 3.1, which contains mostly ambiguous ions.

Figure 4

$E_3$ versus $E_2$ spectrum of events shown in Fig. 3. Region 4.1 was defined to show that the events in this region arise from a high-energy tail of the beam and to define Region 6.1 (see text and Fig. 6).

Figure 5

$E_3$ versus $E_2$ distribution of events in Region 3.1 in Fig. 3. Region 5.1 contains good ${}^{32}\mathrm{Ar}$ ions and has 61 times more events than the surrounding Region 5.2 which contains ambiguous ions.

Figure 6

$E_1$ versus TOF${}_1$ distribution of events in Region 4.1. The correlation between TOF${}_1$ and $E_1$ shows that these events arise from a high-momentum tail of the implanted ion distribution.

Figure 7

Same as Fig. 5, except that events corresponding to sparks in the RPMS (see text) have been rejected.

Figure 8

(Top panel) Singles delayed proton spectrum in ${\mathrm{D}}_3$ for a typical run (histogram) along with the fit to the Monte Carlo simulation (solid line). The filled curves correspond to proton emission following the superallowed transition and the dashed lines are the backgrounds. (Bottom panel) Same as above but all runs added together and gated by $E_{\gamma }=1249$ keV which selects the $p_1$ group at $E_p\approx 2.3$ MeV. In both cases, the ratio of the residuals to the standard deviation for each point is shown below the corresponding spectrum.

Figure 9

Simplified ${}^{32}\mathrm{Ar}$ decay scheme showing observed transitions that produce γ rays.

Figure 10

(Upper panel) ISOLDE data and R-matrix fit. (Lower panel) Ratio of residuals to standard deviation for each point. This spectrum was taken by implanting 60-keV ${}^{32}\mathrm{Ar}$ ions into a $20\hspace{1em}\mu$g/cm${}^2$ carbon foil and observing the β-delayed proton groups with cooled PIN diodes. The detection setup was immersed in a 3.5-T magnetic field that prevented the ${\beta }^{+}$s from reaching the detectors and summing with protons.

Figure 11

Total proton branches (bottom) and background levels (top) deduced from fitting all of the ${\mathrm{D}}_3$ spectra to MC simulations. A sample of one of these fits is given in the top panel of Fig. 8, where we show the result for run 78. The open circles are runs where the contaminant level of ${}^{32}\mathrm{Cl}$ resulted in a background level that was more than 10% above what we would expect based on the decay chain of ${}^{32}\mathrm{Ar}$. For this reason, these runs (above the dashed line in the top plot) were excluded from the fit to $b_{\mathrm{tot}}^{\beta p}$ shown in the bottom plot.

Figure 12

Spectrum of γ rays in coincidence with a β signal in ${\mathrm{D}}_2$, ${\mathrm{D}}_3$, or ${\mathrm{D}}_4$. Lines from ${}^{32}\mathrm{Ar}$ decay are marked with a “$*$”. The remaining labeled lines are from ${}^{32}\mathrm{Cl}$ decay. For all strong lines the corresponding sum peaks (with energy of 511 keV plus the photon energy) were clearly visible.

Figure 13

The absolute ${\epsilon }_{\gamma }{\epsilon }_{\beta }$ from ${}^{32}\mathrm{Cl}$ decays (points) and the normalized PENELOPE curves for each detector (line). The point at 2.2 MeV was not used in normalizing the curves as explained in the text.

Figure 14

(Top panel) Spectrum of γ rays in coincidence with a 3878-keV γ. All of the peaks except for the ones at 2230 and 511 keV correspond to the expected γ cascades following the 3878-keV γ ray. (Bottom panel) Same as above, but with the coincidence gate shifted down by 8 keV. This shows that the 2230-keV γ comes mainly from coincidences with the Compton continuum of the 4770-keV γ ray in ${}^{32}\mathrm{S}$.

Figure 15

The region of interest where the γ ray corresponding to the decay from the $T=2$ state to the ground state should appear at $E_{\gamma }=5046$ keV. The main 5038-keV peak is the first escape from the 5550-keV γ ray from ${}^{32}\mathrm{Cl}$.

Figure 16

The 2836- and 2839-keV γ-ray doublet. The insets show the coincidence proton spectrum with GATE1 (which has a peak corresponding to 610-keV protons) and GATE2, respectively. This shows that the 2836-keV component is followed by proton emission, whereas the 2839-keV γ is not.

Figure 17

$E_4$ versus $E_3$ spectrum of events (roughly calibrated) in Region 3.1 of Fig. 3. In the projections, the data (points) are compared to our Monte Carlo simulation (histogram). Note that the Monte Carlo projections have not been rescaled to fit the data; the normalization is determined from the total number of ${}^{32}\mathrm{Ar}$.

Figure 18

$E_4$ spectrum (roughly calibrated) of events in Region 3.1. This is similar to Fig. 17 but with the extra condition on Region 5.1.

Figure 19

Superallowed proton peak in ${\mathrm{D}}_3$ coincident with ${\mathrm{D}}_2$ (solid line) or ${\mathrm{D}}_4$ (dashed line). The difference in widths is due to the differing solid angles covered by ${\mathrm{D}}_4$ and ${\mathrm{D}}_2$. The curve for the coincidences with ${\mathrm{D}}_4$ has been multiplied by $0.65$ to show more clearly the energy shift. The arrows should coincide if the ${}^{32}\mathrm{Ar}$'s were implanted exactly in the center of ${\mathrm{D}}_3$.