AbstractThe Cryogenic Rare Event Search with Superconducting Thermometers (CRESST) experiment aims at the direct detection of dark matter particles via their elastic scattering off nuclei in a scintillating CaWO$_4$ 4 target crystal. The CaWO$_4$ 4 crystal is operated together with a light detector at mK temperature and read out by a Transition Edge Sensor. For many years, CaWO$_4$ 4 crystals have successfully been produced in-house at Technical University of Munich (TUM) with a focus on high radiopurity which is crucial to reduce background originating from radioactive contamination. In order to further improve the CaWO$_4$ 4 crystals, an extensive chemical purification of the raw materials and the synthesised CaWO$_4$ 4 powder has been performed. In addition, a temperature gradient simulation of the growth process and subsequently an optimisation of the growth furnace with the goal to reduce the intrinsic stress was carried out. We present results on the intrinsic stress in the CaWO$_4$ 4 crystals and on the CaWO$_4$ 4 powder radiopurity. A crystal grown from the purified material was installed in the current CRESST set-up. The detector is equipped with an instrumented holder which is used to measure the alpha decay rate of the crystal. We present a preliminary analysis showing a significantly reduced intrinsic background from natural decay chains.