Critical Casimir force in ${}^4$He films: confirmation of finite-size scaling

We present new capacitance measurements of critical Casimir force-induced thinning of ${}^4$He films near the superfluid/normal transition, focused on the region below $T_{\lambda }$ where the effect is the greatest. ${}^4$He films of 238, 285, and 340 \AA thickness are adsorbed on N-doped silicon substrates with roughness $\approx 8\mathring{A}$. The Casimir force scaling function $\vartheta$, deduced from the thinning of these three films, collapses onto a single universal curve, attaining a minimum $\vartheta =-1.30\pm 0.03$ at $x=t{d}^{1\mathrm{/}\nu }=-9.7\pm 0.8{\mathring{A}}^{1\mathrm{/}\nu }$. The collapse confirms the finite-size scaling origin of the dip in the film thickness. Separately, we also confirm the presence down to $2.13K$ of the Goldstone/surface fluctuation force, which makes the superfluid film $\sim 2\mathring{A}$ thinner than the normal film.