$^4$He Films Adsorbed in Porous MCM--41 Ceramic: Crossover to One-dimensional Superfluidity

The superfluid density of $^4$He films adsorbed in MCM-41 ceramic having long \hbox{40-\AA}-diameter cylindrical pores is studied with a torsion oscillator technique. Finite-size Kosterlitz-Thouless transitions are observed in the data, with the broadening of the transitions increasing considerably as the film thickness decreases. The data are in good agreement with the Machta-Guyer theory of the KT transition in a cylindrical channel, but fits to the broadening require an increasing vortex core size. In the thinnest films a crossover to the one-dimensional behavior predicted theoretically is observed, where the vortex core radius limits at the cylinder radius and the $T = 0$ superfluid density approaches zero.