Fractal surface dimensionality and finite-size scaling at the superfluid transition of confinedHe4

A connection between surface fractal dimensionality and the critical behavior of confined ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ at the superfluid transition is suggested. It is shown that in the specific case of Nuclepore filters the disagreement with bulk-correlation-length scaling can be resolved if one ascribes a fractal dimension to the confining surface of about 2.34. Explicit scaling functions for both the specific heat and the superfluid density are suggested. Measurements of adsorption isotherms for 0.2-μm filters support this fractal conjecture and yield a surface dimensionality of 2.24±0.02.