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We have studied the superfluid transition of 4He in situations where the growth of the correlation length is limited by a uniform spatial confinement. Under these conditions the critical behavior attained in the thermodynamic limit is greatly modified. One expects that data for similar confinement should scale with the correlation length. This is found to be true for the specific heat in planar confinement, 2D crossover, except in the region of the specific heat maximum and on the superfluid side. The modifications due to confinement depend on the details of the geometry. Studies of the specific heat with 1D and 0D crossover show the important role played by the lower dimension. Far from the transition the details of the confining geometry reveal contributions to the specific heat which can be attributed to surfaces and edges. Comparison of experimental resugts with theoretical calculations show agreement in some areas and disagreement in others.

We report studies of a 4He film confined between two silicon wafers separated by 3189 A. The film is connected to a bulk helium reservoir via small channels 100 A high, 8 µm wide by 2000 µm long. This cell design has allowed us to study the heat capacity in a planar confinement (a film of ∞ lateral size), and the superfluid density in the connecting channels (a film of finite lateral size). This work is relevant to finite-size scaling of the specific heat for 2D confinement and it is compared with earlier data. It is also relevant to finite-size 2D behavior for the superfluid density which is related to the recent theory of Sobnack and Kusmartsev. Analysis of the data is presented as well as a discussion of future cell designs to address, in particular, the behavior of laterally confined films.

We report measurements of the specific heat near the superfluid transition of 4He confined in uniform channels of 1 µm square cross-section. This system undergoes a crossover from three dimensional behavior (3D) to 1D as the transition is approached. This resugts in a substantial rounding of the specific heat maximum as well as a shift to colder temperatures relative to the bulk system. We compare these data to previous measurements where crossovers from 3D to 2D and 0D were studied with the smallest confining dimension being the same (1 µm) in each case. We also compare these resugts in the context of finite-size scaling to previous studies where crossover to 1D was measured in cylindrical geometries. We identify regions where surface and edge effects dominate the specific heat, and compare these amplitudes to theory, where available. The realization of the confining geometry in this work is achieved with a combination of silicon lithography and direct wafer bonding.

When helium is confined to a uniform small dimension L, its behavior is modified due to the limitation of the growth of the correlation length. This has been explored for planar confinement over a wide range of L. Less is known about confined mixtures, especially in the case of the specific heat. Here two principal effects come into play: the magnitude of the correlation length ξ increases, and the measured heat capacity Cpx must be converted to Cpφ, where φ=μ3−μ4, before finite-size scaling predictions can be verified. The increase in ξ makes a given confinement L “look smaller” as the concentration x is increased. This, as well as changes of L itself, can be used to test predictions of correlation-length scaling. We report measurements of the specific heat of confined mixtures and compare with the well established scaling for the pure system.

We report an analysis of a superfluid Helmholtz resonance in the case of helium confined in a superleak. The resonance of the superfluid is achieved under nearly adiabatic conditions. Equations are derived for the resonance frequency, the temperature oscillations of the superleak and the phase relation of this signal relative to an ac heat input. The resonance frequency yields the superfluid fraction of the confined helium. Data are analyzed as function of frequency and temperature and yield parameters such as the dissipation and thermal conductivity which determine the resonance line shape. Estimates are made of the thermodynamic parameters in the resonance equation by using derivatives along the pressure-temperature-concentration lambda surface. These parameters are compared with results from the analysis of the resonance.

We report measurements of the specific heat of 4He near the superfluid transition while confined between silicon wafers at 0.9869 μm separation. These data are analyzed to check on the behavior expected from correlation-length scaling. Comparison is also made with other data for planar confinement, as well as data for cylindrical confinement. These represent different lower-dimensional crossovers. We find that the present data scale very well above the bulk transition temperature, and in the region immediately below it. Near the specific heat maximum however, the data for planar confinement do not collapse on a universal curve. We compare these results with specific theoretical scaling functions. In particular we find that on the normal side, and for large enough values of the scaling variable, one can describe the data well using the concept of the surface specific heat. The locus of the data in this region agrees well with the most recent theoretical calculations.

We report heat capacity measurements of confined films of4He. These studies were undertaken to test predictions of correlation-length scaling. They are the first measurements for completely confined films over a range of confinements, and represent a geometry where criticality changes from 3-dimensions (3D) to 2D. The finite system is realized with a4He film confined between two, 2″ diameter, silicon wafers, which are separated by a small gap. A new technique was developed to bond these wafers at a uniform separation. The gap size, which determines the film thickness, ranges from 0.05 to 0.7 μm 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, as in conventional ac calorimetry, but with simultaneous dc regulation of the average temperature. The data are analyzed using a modified Sullivan–Seidel equation, which takes into account in an empirical way the finite conductivity of the cell. This procedure yields heat capacity data with good absolute accuracy and high resolution. Scaling analysis of the data both above and below the bulk transition temperature shows collapse onto universal curves determined only by the ratio of the correlation length to the confinement size. This is true everywhere except near the heat capacity maximum. Here, and into the superfluid side there is lack of scaling which might be associated with 2D crossover. We compare this result with calculations of scaling functions and find that these tend to underestimate the effect of confinement. Comparison with earlier results for cylindrical confinement shows differences which are most striking in the region of the specific heat maximum. The cylindrical and planar confinement data follow similar trends above 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 νeff=0.674±0.001.

We report new specific heat data for4He confined between two silicon wafers at 0.0483 μm separation. This extends our work by a factor of two in the range of confinements studied, and allows us a better check on scaling predictions. The present data have been obtained with a new design for the SiO2pattern which is used to achieve the wafer's separation. With this, we have obtained reliable data in the region where the confined helium becomes superfluid. We find that for T > Tλthe new data collapse well onto a universal function. Below Tλ, but above the specific heat maximum, the data also collapse well. Near the maximum, however, there is lack of collapse which persists into the superfluid region. It seems likely that this is connected with the two-dimensional crossover for the geometry of these cells