Phase transition of two-dimensional ³He from a dilute to a dense phase

We have measured the heat capacity of ³He in films of ⁴He as thin as 10 Å formed on a Nuclepore filter substrate. At low temperatures, where the ³He is in the lowest state as far as motion perpendicular to the film surface, we find that the ³He undergoes a transition from a dilute phase to a dense phase. We have observed this transition for films of ⁴He of 12.3- and 10-Å thickness and for coverages of ³He below about 0.2 atomic layers. For thicker ⁴He films, and higher ³He coverages, the ³He remains homogeneously spread out over the surface of the ⁴He. The striking characteristic of the transition is the sudden onset of linear temperature dependence with a slope which is proportional to the amount of ³He in the calorimeter. This is consistent with the formation of islands of a dense two-dimensional phase which grows in extent proportionately to the amount of ³He. Two puzzling aspects of our results are the lack of a significant heat-capacity jump at the transition and, based on a linear extrapolation of the data to zero temperature, a substantial amount of missing entropy.