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Australian Journal of Physics

Critical Phenomena in Systems of Finite Thickness. II. Ideal Bose Gas

Michael N. Barber

Published Jan 1, 1973• 21 citations
Publisher

Abstract

The grand potential pressure of an ideal boson film which is infinite in d-1 dimensions (d > 2) but of finite thickness L in the dth dimension is analysed in the limit L -+ CX), to order L -d. Across the film (i) periodic, (ii) antiperiodic, and (iii) hard wall boundary conditions are applied. The corresponding shifts in the critical temperature Te.iL) are found to vary, for large L, as (i) L -d+2 for all d ;;:. 3, (ii) L-1, L -2lnL, and L -2 for d = 3,4, and ;;:. 5 respectively, and (iii) L -1lnL and L -1 for d = 3 and ;;:. 4. With hard wall boundary conditions, the critical temperature of a sufficiently thick film exceeds that of the bulk, while the other boundary conditions depress Te. The bulk critical region is asymptotically "rounded" on a scale set by the bulk correlation length matching the thickness L in accord with the scaling theory of finite-size effects.

DOI
10.1071/ph730483

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arXiv:0907.1613•Physical Review E•Jun 2, 2010•20 citations
The grand potential pressure of an ideal boson film which is infinite in d-1 dimensions (d > 2) but of finite thickness L in the dth dimension is analysed in the limit L -+ CX), to order L -d. Across the film (i) periodic, (ii) antiperiodic, and (iii) hard wall boundary conditions are applied. The corresponding shifts in the critical temperature Te.iL) are found to vary, for large L, as (i) L -d+2 for all d ;;:. 3, (ii) L-1, L -2lnL, and L -2 for d = 3,4, and ;;:. 5 respectively, and (iii) L -1lnL and L -1 for d = 3 and ;;:. 4. With hard wall boundary conditions, the critical temperature of a sufficiently thick film exceeds that of the bulk, while the other boundary conditions depress Te. The bulk critical region is asymptotically "rounded" on a scale set by the bulk correlation length matching the thickness L in accord with the scaling theory of finite-size effects.
The specific heat at constant density of an ideal-Bose-fluid film of thickness $L=l{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, but infinite lateral extent, is calculated analytically to order ${l}^{\ensuremath{-}2}$. Both hard-wall and periodic boundary conditions are considered. Good agreement with the numerical calculations of Goble and Trainor for the total specific heat under hard-wall conditions is obtained down to $l\ensuremath{\simeq}10$. In the critical region, the large-lll behavior accords with the scaling theory of finite-size effects. The appropriate scaling functions and the surface specific heat are explicitly calculated.
Critical phenomena in films of finite thickness are considered. A detailed scaling theory, with allowance for distinct exponents $\ensuremath{\lambda}$ and $\ensuremath{\theta}=\frac{1}{\ensuremath{\nu}}$ for the critical-point shift and rounding, respectively, is confirmed by exact calculations on ddd-dimensional ferromagnetic spherical models and ideal Bose fluids with various boundary conditions. Isingmodel results and existing data on real helium films are consistent with the theory.
A detailed analysis of the finite–size effects on the bulk critical behaviour of the d–dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N) and a combination of Neumann and Dirichlet (ND) on both surfaces. A systematic method for the evaluation of the finite–size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite–size scaling form at the bulk critical temperature only for (p) and (a). For the remaining boundary conditions the standard finite–size scaling hypothesis is not valid. At d = 3, the critical amplitude (related to the so called Casimir amplitude) is estimated. We obtain
A detailed analysis of the finite-size effects on the bulk critical behaviour of the d -dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic ( p ), antiperiodic ( a ) and free surfaces with Dirichlet ( D ), Neumann ( N ) and a combination of Neumann and Dirichlet ( ND ) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d . It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for ( p ) and ( a ). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., implying a fluctuation-induced attraction between the surfaces for ( p ) and repulsion in the other two cases. For ( D ) and ( N ) we find a logarithmic dependence on L .
Following the approach of Barber and Fisher, we formulate a finite-size scaling theory for the Bose condensate. Using bulk results as input, we make a number of predictions for the behaviour of the condensate fraction f0(L, T) in an ideal Bose system confined to a hypercube, of side L, in d dimensions. A comparison is made with analytical results for a system in three dimensions under a variety of boundary conditions. While the standard temperature variable t[= (T – Tc)/Tc] is appropriate in the case of periodic and antiperiodic boundary conditions, the use of a shifted variable t[= t – ε(L), where ε(L) = O(L−1 In L)] is essential in the case of Neumann and Dirichlet boundary conditions. Nonetheless, in each case, the predictions of the scaling formulation are fully borne out. Finally, the formulation is extended (i) to include the so-called surface condensate, and (ii) to cover all temperature down to 0 K.
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature Tc. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension d*=3 , where the critical exponent 1-α-ν=(d-3)∕2 of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above Tc for 1<d<4 because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature Tc,film(L)<Tc exists at finite film thickness L . Our results include an exact description of the dimensional crossover between the d -dimensional finite-size critical behavior near bulk Tc and the (d-1) -dimensional critical behavior near Tc,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region Tc,film(L)≤T≤Tc. For 2<d<4 , the Gaussian results are renormalized and reformulated as one-loop contributions of the φ4 field theory at fixed dimension d and are then compared with the ε=4-d expansion results at ε=1 as well as with d=3 Monte Carlo data. For d=2 , the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the d -dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.