We study the change of the size and structure of freely suspended single lamella nanoparticles of polyethylene during thermal annealing in aqueous solutions. Using small-angle X-ray scattering and cryogenic transmission electron microscopy, it is shown that a doubling of the crystalline lamella sandwiched between two amorphous polymer layers is obtained by annealing the nanoparticles at 125 degrees C. This thickening of the crystalline lamella can be understood in terms of an unlooping of polymer chains within a single nanoparticle. In addition, a variation of the annealing temperature from 90 to 115 degrees C demonstrates that the inverse of the crystalline lamellar thickness increases linearly with the annealing temperatures leading to a recrystallization line in a Gibbs Thomson graph. Since the nanoparticles consist of about only eight polymer chains, they can be considered as a ideal candidates for the experimental realization of equilibrium polymer crystals.