Characterisation of low background CaWO$_4$ crystals for CRESST-III

The CRESST-III experiment aims at the direct detection of dark matter particles via their elastic scattering off nuclei in a scintillating CaWO$_4$ target crystal. For many years CaWO$_4$ crystals have successfully been produced in-house at Technische Universität München with a focus on high radiopurity. To further improve the CaWO$_4$ crystals, an extensive chemical purification of the raw materials has been performed and the crystal TUM93 was produced from this powder. We present results from an $\alpha$-decay rate analysis performed on 344 days of data collected in the ongoing CRESST-III data-taking campaign. The $\alpha$-decay rate could significantly be reduced.