Precise Half Life Measurement of $^{10}$C

We have measured the half-life of $^{10}$C as part of our program to test the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix \textit{via} $0^+\to 0^+$superallowed $\beta $ transitions. The $^{10}$C half-life has been measured twice before, with precisions of 0.10{\%} and 0.08{\%}. With our current techniques, we expect to be able to improve that precision by a factor of two. To obtain $^{10}$C, we used a $^{11}$B primary beam at 23$A~$MeV to bombard a cryogenic hydrogen target. From the reaction products, a high purity $^{10}$C beam of 18.5$A~$MeV was produced by the MARS spectrograph. The beam was then extracted in air, passed through a 0.3-mm-thick BC-404 plastic scintillator and a set of Al degraders, which had been adjusted so that the $^{10}$C nuclei stopped in the center of the 76-$\mu $m-thick aluminized-mylar tape of our fast tape-transport system. We collected $^{10}$C nuclei for 10, 15 or 20~s; then the beam was switched off and the activity was moved in less than 0.2~s to the center of a 4$\pi $ proportional counter, located in a well shielded region. The observed decays were then multi-scaled over a 400~s time span. To ensure an unbiased result, we split the experiment into several runs, each differing from the others in its discriminator threshold, detector bias or dominant dead-time setting. The analysis of these separate runs showed no systematic bias with these parameters. Our preliminary result is t$_{1/2}(^{10}$C) = 19.313(10)~s, a result with 0.05{\%} precision.