Fine-Structure Constant Connects the Polarizability of Atoms and Vacuum

Abstract

We examine the recently derived quantum-mechanical relation between atomic polarizabilities and equilibrium internuclear distances in van der Waals (vdW) bonded diatomic systems [Phys. Rev. Lett. {\bf 121}, 183401 (2018)]. For homonuclear dimers, this relation is described by the compact formula $α_{\rm m}^{\rm q} = ΦR_{\rm vdW}^7$, where the constant factor in front of the vdW radius was determined empirically. Here, we derive $Φ= (4πε_0/a_0^4) \times α^{4/3}$ expressed in terms of the vacuum electric permittivity $ε_0$, the Bohr radius $a_0$, and the fine-structure constant $α$. The validity of the obtained formula is confirmed by estimating the value of the fine-structure constant from non-relativistic quantum-mechanical calculations of atomic polarizabilities and equilibrium internuclear vdW distances. The presented derivation allows to interpret the fine-structure constant as the ratio between the polarizability densities of vacuum and matter, whereas the vdW radius becomes a geometrical length scale of atoms endowed by the vacuum field.