$2\nu\beta\beta$ of $^{96}$Zr to the First Excited $0^+$ State

A progress report is presented on our work to measure the double-beta decay of $^{96}$Zr to the first excited $0^+$ state of $^{96}$Mo. Such measurements provide valuable test cases for $2\nu\beta\beta$ nuclear matrix element calculations, which in turn are used to tune $0\nu\beta\beta$ nuclear matrix element calculations. After undergoing double-beta decay, the excited $0^+$ state decays via the $0^+\rightarrow 2^+ \rightarrow 0^+$ decay sequence in the daughter nucleus, emitting two coincident $\gamma$ rays. These two $\gamma$ rays are detected in coincidence by two HPGe detectors sandwiching the $^{96}$Zr sample, with a NaI veto in anti-coincidence. This experimental apparatus, located at the Kimballton Underground Research Facility (KURF), has previously been used to measure the $T_{1/2}$ of $^{100}$Mo and $^{150}$Nd to the first excited $0^+$ states. The present experiment is an attempt to detect this decay mode in a third nuclide. The experiment is hindered by our small sample mass of 17.9 grams of enriched $^{96}$Zr, which has a natural abundance 2.8$\%$. Preliminary results will be shown.