High-precision half-life determination for ${}^{21}\mathbf{Na}$ using a $4\pi$ gas-proportional counter

A high-precision half-life measurement for the superallowed ${\beta }^{+}$ transition between the isospin $T=1/2$ mirror nuclei ${}^{21}\mathrm{Na}$ and ${}^{21}\mathrm{Ne}$ has been performed at the TRIUMF-ISAC radioactive ion beam facility yielding $T_{1/2}=22.4506\left(33\right)$ s, a result that is a factor of 4 more precise than the previous world-average half-life for ${}^{21}\mathrm{Na}$ and represents the single most precisely determined half-life for a transition between mirror nuclei to date. The contribution to the uncertainty in the ${}^{21}\mathrm{Na} \mathcal{F}{t}^{\mathrm{mirror}}$ value due to the half-life is now reduced to the level of the nuclear-structure-dependent theoretical corrections, leaving the branching ratio as the dominant experimental uncertainty.

Figure 1

Voltage plateau measurements for the $4\pi$ gas counter at the highest (black circles) and lowest (red squares) discriminator settings used in this experiment, with lines connecting the data points drawn to aid the eye. The voltage range over which the detector was operated for the ${}^{21}\mathrm{Na}$ half-life measurements is bracketed by vertical dashed lines and shown in the inset.

Figure 2

$\beta$-coincident $\gamma$-ray spectrum summed for all ${}^{21}\mathrm{Na}$ half-life runs, with the origin of the $\gamma$-ray peaks labeled. No transitions from nuclei other than ${}^{21}\mathrm{Na}$ and room background are identifiable.

Figure 3

Decay data from a typical run composed of five cycles, with fit overlaid and residuals (bottom). The mean and sigma of the residuals, and the ${\chi }^2/\nu$ for the fit, are consistent with normally distributed data.

Figure 4

Deduced ${}^{21}\mathrm{Na}$ half-lives for each experimental run, with the weighted average and $\pm 1{\sigma }_{\mathrm{stat}}$ limits indicated by the solid and dashed red lines, respectively. The ${\chi }^2/\nu$ of 1.02 indicates excellent statistical consistency amongst the resulting half-lives.

Figure 5

Weighted averages of the ${}^{21}\mathrm{Na}$ half-life values obtained from fits to the cycles with leading channels removed. Resulting half-lives were averaged when the ${}^{21}\mathrm{Na}$ activity at the beginning of the fit was equal to within $\pm 200$ Hz of the value indicated on the $x$ axis. The data points are strongly correlated since individual cycles contribute to multiple data points, however, no cycle contributes more than once to any individual data point.

Figure 6

The deduced cycle-by-cycle ${}^{21}\mathrm{Na}$ half-lives plotted as a function of the measured initial ${}^{21}\mathrm{Na}$ activity. The slope obtained from a linear regression yields $5.8\pm 11.6\times {10}^{-7}$ s/Hz, a value consistent with zero indicating no systematic associated with the initial rate in the gas counter.

Figure 7

Half-life measurements grouped according to experimental parameters. See text for details.

Figure 8

Box plots for the run-by-run half-lives measured with the new MCS grouped according to the dwell-time parameter. The horizontal line through each box is the median of the distribution, while the vertical extent of the box encompasses the interquartile range (25% to 75% of the distribution). The error bars extend to 1.5 times the interquartile range, after which data is rendered as discreet points. The width of each box is proportional to the square root of the number of runs measured at that dwell setting, while the notches represent the 95% confidence interval of the median.

Figure 9

World-average half-life of ${}^{21}\mathrm{Na}$. Shown are measurements from Refs. [7] (Al74), [8] (Az75), [6] (Gr15), and this paper. The result from Ref. [19] has been omitted as it is $\sim 60$ times less precise than this paper, and thus has no impact on either the average half-life or its uncertainty. See text for further discussion.

Figure 10

Fractional uncertainties in the $\mathcal{F}{t}^{\mathrm{mirror}}$ value for ${}^{21}\mathrm{Na}$ based on the $Q_{EC}$ value, branching ratio (BR), and half-life from Refs. [20] and [6], the theoretical corrections from Ref. [4], and the half-life value reported here. The different half-life contributions represent the averaging scenarios described in the text. The contribution from the half-life to the $\mathcal{F}{t}^{\mathrm{mirror}}$ error budget is now comparable to that from the nuclear-structure-dependent corrections if the new (inflated) world average is adopted, and insignificant if the high-precision $T_{1/2}$ result from the present paper is adopted.