Thermal coupling of silicon oscillators in cryogen-free dilution refrigerators.

Silicon double paddle osillators (DPO) have been successfully used for measuring the elastic properties of amorphous films down to 10 mK (see for example \cite{Fefferman16,Liu14}). Until now, our group has used a wet dilution refrigerator for the lowest temperature measurements. We present measurements carried out on a Bluefors cryogen-free dilution refrigerator that demonstrate an extreme sensitivity of the thermal coupling of the DPO to its environment. These measurements show that it is necessary to enclose the DPO in a shield at the mixing chamber (MXC) temperature. Any gaps in the shield limit its effectiveness, even if there is no line-of-sight path to the DPO. In the absence of a cryogenic hermetic shield surrounding the DPO, turning off the pulse tube while maintaining the MXC and still temperatures leads to heating of the DPO. This demonstrates that any heating of the sample due to pulse tube vibrations is a less important effect.