  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.
