Low-temperature thermal conductivity measurements of Al$_2$O$_3$ ceramic for use in bolometric particle detector*

Bolometric particle detectors for rare weak processes operate at temperatures as low as 10mK and are background-dependent, so radiopure structural materials such as alumina ceramic (Al$_2$O$_3$) are of interest, and their thermal properties in the very low temperature regime must be understood. Our experiments are conducted in a dilution refrigerator, with heaters being used to create temperature gradients across elongated alumina samples of different cross-sectional geometries mounted in copper clamps, with one end thermalized on the 10mK plate of the cryostat. Temperatures of both ends are measured with RuO$_2$ resistance thermometers, and thermal conductivity k(T) can be determined using the relationship $ \frac{dQ}{dt} =\frac{A}{l}{\int{k(T)dT}}$, where $\frac{dQ}{dt}$ is heating power, $A$ is cross-sectional area of the sample, $l$ is its length, and $T$ is temperature. Absolute values and temperature dependence of thermal conductivity of the alumina samples were measured and compared to well-investigated single-crystal sapphire properties. Thermal conductivity of other materials of interest was also investigated; the results will be presented.