Anomalous effective dimensionality of quantum gas adsorption near nanopores

Three problems involving quasi-one-dimensional (1D) ideal gases are discussed. The simplest problem involves quantum particles localized within the groove, a quasi-1D region created by two adjacent, identical and parallel nanotubes. At low temperature (T), the transverse motion of the adsorbed gas, in the plane perpendicular to the axes of the tubes, is frozen out. Then, the low T heat capacity C(T) of N particles is that of a ID classical gas: C*(T) = C(T)/(Nk_B) → 1/2. The dimensionless heat capacity C*increases when T ≥ 0.1T_x,y (transverse excitation temperatures), asymptoting at C*= 2.5. The second problem involves a gas localized between two nearly parallel, co-planar nanotubes, with small divergence half-angle y. In this case, too, the transverse motion does not contribute to C(T) at low T, leaving a problem of a gas of particles in a ID harmonic potential (along the z axis, midway between the tubes). Setting ω_z as the angular frequency of this motion, for T ≥ τ_z = ω_zh/k_B, the behavior approaches that of a 2D classical gas, C*= 1; one might have expected instead C*= 1/2, as in the groove problem, since the limit γ ≡ 0 is 1D. For T ≪ C τ_z, the thermal behavior is exponentially activated, C*∼ (τ_z/T)²e ^-τz/T. At higher T (T ≈ ∈_y/k_B ≡ τ_y ≫ τ_z), motion is excited in the y direction, perpendicular to the plane of nanotubes, resulting in thermal behavior (C*= 7/4) corresponding to a gas in 7/2 dimensions, while at very high T (T > hω_x/k_B ≡ τ_x ≫ τ_y), the behavior becomes that of a D = 11/2 system. The third problem is that of a gas of particles, e.g. ⁴He, confined in the interstitial region between four square parallel pores. The low T behavior found in this case is again surprising - that of a 5D gas.