Thermal similarity in two and three dimensions: The Debye temperatures of bulk and adsorbed monolayerHe4

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The Debye temperatures for two- and three-dimensional systems are compared using an isotropic elastic model that includes the effect of finite initial stress and assumes the Cauchy conditions. It is shown that the effect of initial pressure on sound speeds, Debye temperatures, and selected elastic properties is determined in both dimensions by the dimensionless product of pressure and compressibility. The model is applied to the observed similarity in Debye temperatures for monolayer and hcp bulk ${\mathrm{He}}^{4}$. Using 0 \ifmmode^\circ\else\textdegree\fi{}K compressibility and pressure data, there is essential agreement between the elastic model and the experimental bulk Debye temperatures. The agreement results from the effect of initial pressure, which softens the bulk Debye temperatures by as much as 20% relative to the zero-pressure values for the same compressibility. A large softening also occurs in the two-dimensional extension when applied to the ${\mathrm{He}}^{4}$-graphite adsorption system but, in contrast to the bulk, the two-dimensional spreading pressures are insufficient to produce comparable quantitative agreement with the experimental film Debye temperatures. The idealized model is extended to include a specific substrate effect by separating the total film elastic constants into two parts: An intrinsic (Cauchy) contribution associated with the two-dimensional adsorbate lattice and a secondary contribution identified with the substrate, which lifts the Cauchy conditions on the total elastic constants for the film. The required fractional substrate contribution to the bulk and shear moduli yielding agreement with the experimental film Debye temperatures is examined.