Surface microparticles in liquid helium. Quantum archimedes' principle

Deviations from Archimedes' principle for spherical molecular hydrogen particles with the radius $R_0$ at the surface of $^4$He liquid helium have been investigated. The classical Archimedes' principle holds if $R_0$ is larger than the helium capillary length $L_{\mathrm{cap}}\cong 500$ $\mu\mathrm{m}$. In this case, the elevation of a particle above the liquid is $h_+\sim R_0$. At $30\,\mu\mathrm{m}<R_0<500\,\mu\mathrm{m}$, the buoyancy is suppressed by the surface tension and $h_+\sim R_0^3/L^2_{\mathrm{cap}}$. At $R_0<30\,\mu\mathrm{m}$, the particle is situated beneath the surface of the liquid. In this case, the buoyancy competes with the Casimir force, which repels the particle from the surface deep into the liquid. The distance of the particle to the surface is $h_-\sim R^{5/3}_c/R^{2/3}_0$ if $R_0> R_c$. Here, $R_{\text{с}}\approx\left(\frac{\hbar c}{\rho g}\right)^{1/5}\approx1 $, where $\hbar$ is Planck's constant, $c$ is the speed of light, $g$ is the acceleration due to gravity, and $\rho$ is the mass density of helium. For very small particles ($R_0<R_c$), the distance $h_-$ to the surface of the liquid is independent of their size, $h_- = R_c$.