Surface Microparticles in Liquid Helium. Quantum Archimedes’ Principle

Deviations from Archimedes’ principle for spherical molecular hydrogen particles with the radius R0 at the surface of 4He liquid helium have been investigated. The classical Archimedes’ principle holds if R0 is larger than the helium capillary length Lcap ≅ 500 μm. In this case, the elevation of a particle above the liquid is h+ ~ R0. At 30 μm < R0 < 500 μm, the buoyancy is suppressed by the surface tension and h+ ~ R30/L2cap. At R0 < 30 μm, 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- ~ R5/3c/R2/30 if R0 > Rc. Here, Rc≅(ℏcρg)1/5≈1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R_c} \cong {\left( {\frac{{\hbar c}}{{\rho g}}} \right)^{1/5}} \approx 1$$\end{document}, where ħ is Planck’s constant, c is the speed of light, g is the acceleration due to gravity, and ρ is the mass density of helium. For very small particles (R0 < Rc), the distance h_ to the surface of the liquid is independent of their size, h_ = Rc.