Mark O. Kimball

Behavior of 4He Near Tλ in Films of Infinite and Finite Lateral Extent

We report studies of a 4He film confined between two silicon wafers separated by 3189 Å. The film is connected to a bulk helium reservoir via small channels 100 Å 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.

1D Crossover, universality and finite-size scaling of the specific heat

We report measurements of the specific heat of 3He-4He mixtures near the superfluid transition when confined to channels of 1 /tm square cross section. These data test the universality of finite-size scaling as function of 3He concentration for 1D crossover. The analysis of these data requires that data measured at fixed concentration be converted to a specific heat at constant chemical potential difference = μ3 - μ4. This is carried out according to a procedure performed for planar mixtures by Kimball and Gasparini. We find that, in the most self-consistent analysis of the data, the mixtures define a separate scaling locus from that of pure 4He, both above and below Tλ. An analysis whereby the exponent a is forced to have the same universal value—as opposed to the best-fit value—yields a good collapse of all the data. This is achieved, however, at a cost of self-consistency. These results mirror very closely those obtained for finite-size scaling of confined planar mixtures, i.e. for 2D crossover.

Specific Heat of Helium in 2 μm3 Boxes, Coupled or Uncoupled?

We report on recent measurements of the specific heat of helium confined in pill‐boxes 2 μm across and 2 μm deep made lithographically on a silicon wafer. The experimental cells distribute liquid from a bulk reservoir to ∼ 108 boxes by an array of very shallow fill‐channels (0.019 μm and 0.010 μm) which represent a negligible volume compared to that of the boxes. Since the channels are so shallow, the helium in them becomes superfluid at a much lower temperature than the liquid in the boxes. Therefore, during the course of the heat capacity measurements, the liquid in the channels in always normal, and the cell would be expected to behave as a system of uncoupled boxes. We compare these measurements with one previously made of a cell where the confinement was to 1 μm boxes with an equivalent fill arrangement. While the shift in the position of the specific heat maximum relative to the 1 μm cell is what one would expect on the basis of finite‐size scaling, there are discrepancies in the specific heat ampli...

Testing The Universality Of The Lambda Transition Using Confined Helium Mixtures

The universality of phase transitions is an important prediction of theories of critical behavior. Simply stated, microscopically different systems near a critical point may be described by universal quantities if their dimensionality is the same and the order parameter has the same degrees of freedom. One way to test this idea is to measure the thermodynamic response of a set of systems to an input where the response changes with a variation in some quantity like spatial confinement, impurity concentration, or even pressure. While the response of each system is different, the behavior may still be described by a common critical exponent if the idea of universality is correct. Confined mixtures of 3He‐4He and pure 4He are believed to satisfy these requirements. Here, amplitudes such as the magnitude of the correlation length and the temperature of the transition both depend upon the concentration of the mixture and the extent of confinement. However, universality predicts the value of the critical exponen...

Determination Of The Bulk Helium Critical Exponents Using Confined Helium

The specific heat of helium homogeneously confined in one or more dimensions is expected to collapse onto a scaling function which depends only on the ratio of the smallest dimension of confinement to the correlation length, written as L/ξ. This may be rewritten to explicitly show the temperature dependence of the correlation length as L/ξ0t−ν, where the constant ξ0 is the prefactor of the correlation length, t is a dimensionless temperature difference from the superfluid transition, and ν is the critical exponent associated with the correlation length. Thus, in principle, one should be able to obtain the exponent ν from the scaling of thermodynamic measurements of confined helium for various L’s. This would represent an independent determination of ν distinct from what is obtained using the behavior of the bulk superfluid density, or via the bulk specific heat and the hyperscaling relation. In practice, this analysis is hampered by the lack of a theoretical expression for the scaling function. We present...

Specific Heat of 4He Confined in Channels of 1 µm Square Cross-Section

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.

4He confined to 1 μm3 boxes, 0D crossover, surface and edge effects

Abstract We report measurements of the specific heat near the superfluid transition of 4 He confined to 1 μm 3 cylindrical boxes patterned in SiO2. This system crosses from a 3D behavior to a 0D behavior near the transition. This has a marked effect on the specific heat as seen by a pronounced rounding of the maximum and a shift to a temperature much lower than the transition of the bulk system (and systems with 2D or 1D crossover). We plot the data according to correlation-length scaling theory and compare this to a planar system with the same smallest confinement. Compared to our previous studies of planar systems, the 0D cell has 3× the surface to volume ratio as well as ∼750× as much edge length. We examine the regions where surface and edge effect contributions can be separated. We find that the data do not reach the expected value for the surface region. There is also evidence for a region where the term associated with edge contributions dominates.

Heat capacity of mixtures of 3He-4He confined to coupled 1 μm boxes

Abstract We have measured the heat capacity of helium mixtures confined to lithographically created cylindrical boxes whose height, 1.08 μm , equals their diameter. The specific heat at constant concentration x, C px , is renormalized due to the 3 He impurity. Thus, in order to observe critical behavior, a conversion must be made to a specific heat at constant φ=μ 3 −μ 4 , C pφ . The confined systems specific heat, near its maximum, has values which rise above the bulk systems specific heat at the same temperature. This is unexpected and is true for both Cpx and Cpφ. This enhancement, not observed with pure 4 He , becomes more dramatic as x and the correlation length increase. The shift of the maximum with x is much larger for the boxes than for 2D confined mixtures. These observations might be related to a “band structure” effect associated with the 18.5 nm channels which connect the cylinders.