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Physical Review A

Bose-Einstein Condensation in Thin Films

Raj K. Pathria

Published Mar 1, 1972• 44 citations
Publisher

Abstract

An asymptotic evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry ($\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}D$) is carried out, under a variety of boundary conditions, without converting summations into integrations. The theoretical results for the Dirichlet boundary conditions (${\ensuremath{\psi}}_{S}=0$) contain all the significant features of the numerical results obtained earlier by Goble and Trainor; in particular, we reproduce the characteristic length ${D}^{*}$ at which the specific heat of the system is at an absolute maximum. It turns out that ${D}^{*}$ is directly proportional to the mean interparticle distance $\overline{l}$, with a constant of proportionality $c(\ensuremath{\approx}20\ensuremath{-}30)$. No such characteristic length appears if boundary conditions other than Dirichlet's are employed. The shift, and the rounding off, of the specific-heat maximum are also studied, and the distinguishing influence of the boundary conditions examined.

DOI
10.1103/physreva.5.1451

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Physical Review A•Aug 1, 1974•10 citations
An asymptotic evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry (\ensuremath∞\ifmmode×\else\texttimes\fi\ensuremath∞\ifmmode×\else\texttimes\fiD\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}D\ensuremath∞\ifmmode×\else\texttimes\fi\ensuremath∞\ifmmode×\else\texttimes\fiD) is carried out, under a variety of boundary conditions, without converting summations into integrations. The theoretical results for the Dirichlet boundary conditions (\ensuremathψS=0{\ensuremath{\psi}}_{S}=0\ensuremathψS​=0) contain all the significant features of the numerical results obtained earlier by Goble and Trainor; in particular, we reproduce the characteristic length D∗{D}^{*}D∗ at which the specific heat of the system is at an absolute maximum. It turns out that D∗{D}^{*}D∗ is directly proportional to the mean interparticle distance l‾\overline{l}l, with a constant of proportionality c(\ensuremath≈20\ensuremath−30)c(\ensuremath{\approx}20\ensuremath{-}30)c(\ensuremath≈20\ensuremath−30). No such characteristic length appears if boundary conditions other than Dirichlet's are employed. The shift, and the rounding off, of the specific-heat maximum are also studied, and the distinguishing influence of the boundary conditions examined.
Our previous evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry is extended to second order in (l‾D\frac{\overline{l}}{D}Dl​), where l‾\overline{l}l is the mean interparticle distance. Explicit expressions, involving a dimensionless parameter yyy, are obtained both for the specific heat of the system and for the temperature at which the specific heat is maximum. Occurrence of the parameter yyy, which is a measure of the chemical potential of the system in terms of the spacing of the single-particle energy levels, enables us to cast our expression for the specific heat of the system into a form consistent with the scaling theory of Fisher and Barber. Our results for the specific-heat maximum, as a function of (Dl‾\frac{D}{\overline{l}}lD​), are a considerable improvement over the first-order ones reported earlier, and have come remarkably close to the ones obtained numerically by Goble and Trainor.
The recent observation by Hasan and Goble of a discrepancy between their numerical results on the specific heat of an ideal Bose gas confined to a thin-film geometry and the earlier analytic calculations of Pathria employing Neumann boundary conditions is shown to be unjustified. In those cases where a true comparison is possible, e.g., under Dirichlet or periodic boundary conditions, the observed deviations turn out to be insignificant if one uses the more recent second-order calculations of Greenspoon and Pathria rather than the original first-order ones of Pathria. For comparison under Neumann boundary conditions, fresh numerical calculations are required. We also present here second-order analytic calculations of the specific heat of this system (i) under mixed boundary conditions (Dirichlet on one wall, Neumann on the other) and (ii) under antiperiodic boundary conditions, and compare them with the relevant numerical calculations of Hasan and Goble.
An asymptotic evaluation of the specific heat of an ideal relativistic Bose gas confined to a cuboidal enclosure (${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}$) is carried out, under periodic boundary conditions, taking into account the possibility of particle-antiparticle pair production in the system. Finite-size corrections to the standard bulk behavior are calculated explicitly in the regions t>0t>0t>0 and t<0t<0t<0, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, such that $|t|\ensuremath{\ll}1$ and $|{L}_{i}t|\ensuremath{\gg}1$. While for t>0t>0t>0 finite-size corrections turn out to be exponential for all geometries, for t<0t<0t<0 this is true only in the case of a film; for other geometries, such as a cuboid or a rectangular channel, these corrections conform to a power law instead. Finally, we consider the situation in the core region, where ∣Lit∣=O(1)|{L}_{i}t|=O(1)∣Li​t∣=O(1), and examine the location t∗{t}^{*}t∗ and the height ${c}_{\ensuremath{\rho}}^{*}$ of the specific-heat maximum; finite-size corrections in this region turn out to be $O({L}_{<}^{\ensuremath{-}1})$, where L<{L}_{<}L<​ denotes the shortest side of the enclosure.
Using techniques developed by Greenspoon and Pathria, and by Chaba and Pathria, a rigorous asymptotic analysis of the onset of Bose-Einstein condensation in a finite three-dimensional system at constant pressure, under Dirichlet boundary conditions, is carried out. Finite size corrections arising from a modification of the density of states of the system, as well as from the discreteness of the single-particle energy levels, play an important role in the critical region. At the critical temperature Tc{T}_{c}Tc​, the volume of the system changes abruptly from being proportional to the total number of particles NNN to being proportional to N34{N}^{\frac{3}{4}}N43​, the number of particles in the condensate rises sharply from $\ensuremath{\propto}{N}^{\frac{1}{4}}$ to $\ensuremath{\propto}N$, and the specific heat CP{C}_{P}CP​ turns out to be proportional to N98{N}^{\frac{9}{8}}N89​. At somewhat lower temperatures, the finite size effects arising from the discreteness of the single-particle energy levels become dominant and the volume of the system, together with CP{C}_{P}CP​, becomes $\ensuremath{\propto}{N}^{\frac{3}{5}}$.
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.
The specific heat of an ideal semi-infinite ($\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}D$) Bose gas is calculated numerically using a number of boundary conditions. The results provide a suitable comparison with the analytical calculations of Fisher and Barber and of Pathria. The agreement is very good except for the Neumann boundary condition where the numerical result shows deviation from that of Pathria. However, the earlier assumption of the existence of a "correlation length" appears to have little physical basis. For the hard-wall case, the anomaly in the specific heat is probably due to constraint on particle density. The distinctive effect of such a boundary condition (${\ensuremath{\Psi}}_{s}=0$) is explored more thoroughly in maximal assemblies ($L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$) where its effect on specific heat shows up prominently.