Superfluid transition of He-4 for two-dimensional crossover, heat capacity, and finite-size scaling

We report heat capacity measurements of confined films of He-4. These studi es were undertaken to test predictions of correlation-length scaling. They are the first measurements for completely confined films over a range of co nfinements, and represent a geometry where criticality changes from 3-dimen sions (3D) to 2D. The finite system is realized with a He-4 film confined b etween two, 2" diameter, silicon wafers, which are separated by a small gap . A new technique was developed to bond these wafers at a uniform separatio n. The gap size, which determines the film thickness, ranges from 0.05 to 0 .7 mu m in the present work, and has better than 1% uniformity. The bonded cells are used to conduct high precision heat capacity measurements using a modified ac technique. This involves oscillating the sample temperature, a s in conventional ac calorimetry, but with simultaneous dc regulation of th e average temperature. The data are analyzed using a modified Sullivan-Seid el equation, which takes into account in an empirical way the finite conduc tivity of the cell. This procedure yields heat capacity data with good abso lute accuracy and high resolution. Scaling analysis of the data both above and below the bulk transition temperature shows collapse onto universal cur ves determined only by the ratio of the correlation length to the confineme nt size. This is true everywhere except near the heat capacity maximum. Her e, and into the superfluid side there is a lack of scaling which might be a ssociated with 2D crossover. We compare this result with calculations of sc aling functions and find that these tend to underestimate the effect of con finement. Comparision with earlier results for cylindrical confinement show s differences which are most striking in the region of the specific heat ma ximum. The cylindrical and planar confinement data follow similar trends ab ove the superfluid transition of bulk helium. Below the transition, however , the present data show much more structure. Fits of the scaled planar data above the transition to an empirical scaling function yield a correlation length exponent of nu(eff)=0.674 +/- 0.001.