Specific heat of helium in 2 μm³ boxes, coupled or uncoupled?

We report on recent measurements of the specific heat of helium confined in pill-boxes 2 μm across and 2 μm deep made lithographically on a silicon wafer. The experimental cells distribute liquid from a bulk reservoir to ∼ 10⁸ boxes by an array of very shallow fill-channels (0.019 μm and 0.010 μm) which represent a negligible volume compared to that of the boxes. Since the channels are so shallow, the helium in them becomes superfluid at a much lower temperature than the liquid in the boxes. Therefore, during the course of the heat capacity measurements, the liquid in the channels in always normal, and the cell would be expected to behave as a system of uncoupled boxes. We compare these measurements with one previously made of a cell where the confinement was to 1 μm boxes with an equivalent fill arrangement. While the shift in the position of the specific heat maximum relative to the 1 μm cell is what one would expect on the basis of finite-size scaling, there are discrepancies in the specific heat amplitude between the 2 μm cell utilizing different depth fill-channels, and with the 1 μm cell. It is possible that the channels, even though normal and of negligible volume, provide a weak coupling between the boxes leading to a collective rather than single-box behavior.