Exact solution of Ginzburg's $Ψ$-theory for the Casimir force in $^4$He superfluid films

Abstract

We present an analytical solution of the Ginzburg's $Ψ$-theory for the behavior of the Casimir force in a film of $^4$He in equilibrium with its vapor near the superfluid transition point, and we revisit the corresponding experiments in light of our findings. We find reasonably good agreement between the $Ψ$-theory predictions and the experimental data. Our calculated force is attractive, and the largest absolute value of the scaling function is $1.848$, while experiment yields $1.30$. The position of the extremum is predicted to be at $x=(L/ξ_0)(T/T_λ-1)^{1/ν}=π$, while experiment is consistent with $x=3.8$. Here $L$ is the thickness of the film, $T_λ$ is the bulk critical temperature and $ξ_0$ is the correlation length amplitude of the system for $T>T_λ$.