Measurement of $\beta$-particles to determine cross sections relevant to the weak r-process

The neutron-rich isotopes with 30 $\leq$ Z $\leq$ 45 are thought to be synthesised in neutrino-driven winds after the collapse of a massive star. This nucleosynthesis scenario, called the weak r-process, is studied using nuclear reaction network calculations. The accuracy of the nucleosynthesis simulations is strongly influenced by the reliability of the nuclear physics input parameters used. Recently, it has been demonstrated that ($\alpha$,n) reactions play a particularly important role in the weak r-process, but their rates -- computed from the cross sections -- are only known with large uncertainties in the astrophysically relevant temperature range. The half-lives of the products of some key reactions are such that, in principle, the cross sections can be studied using the activation technique. In many cases, however, the $\beta$-decay of the reaction products leads to the ground state of the daughter nucleus, hence no gamma emission occurs. The purpose of this manuscript is to present our setup with which we determine the cross sections by measuring the yield of $\beta$-particles emitted during radioactive decay. The $^{86}$Kr($\alpha$,n)$^{89}$Sr reaction cross-section measurement is used, as a case study.