The superfluid transition in helium clusters

Abstract

We address cluster size effects on the λ temperature (Tλ) for the rounded-off transition for the Bose–Einstein condensation and for the onset of superfluidity in (4He)N clusters of radius R0=aN1/3, where a=3.5 Å is the constituent radius. The phenomenological Ginsburg–Pitaevskii–Sobaynin theory for the order parameter of the second-order phase transition, in conjunction with the free-surface boundary condition, results in a scaling law for the cluster size dependence of Tλ, which is defined by the maximum of the specific heat and/or from the onset of the finite fraction of the superfluid density. This size scaling law (Tλ0−Tλ)/Tλ0∝R0−1/ν∝N−1/3ν, where ν (=0.67) is the critical exponent for the superfluid fraction and for the correlation length for superfluidity in the infinite bulk system, implies the depression of the finite system Tλ relative to the bulk value of Tλ0. The quantum path integral molecular dynamics simulations of Sindzingre, Ceperley, and Klein [Phys. Rev. Lett. 63, 1601 (1989)] for N=64, 128, together with experimental data for specific heat of He4 in porous gold and in other confined systems [J. Yoon and M. H. W. Chan, Phys. Rev. Lett. 78, 4801 (1997); G. M. Zahssenhaus and J. D. Reppy, ibid. 83, 4800 (1999)], are accounted for in terms of the cluster size scaling theory (Tλ0−Tλ)/Tλ0=(πξ0/a)3/2N−1/2, where ξ0=1.7±0.3 Å is the “critical” amplitude for the correlation length in the bulk. The phenomenological theory relates Tλ for the finite system to the correlation length ξ(T) for superfluidity in the infinite bulk system, with the shift (Tλ0−Tλ) being determined by the ratio R0/ξ(T), in accord with the theory of finite-size scaling.