The Einstein nanocrystal

Abstract

We study the simplest possible model of nanocrystal consisting in a simple cubic lattice with a small number of atoms ($N_A \sim 10-10^3$), where each atom is linked to its nearest neighbor by a quantum harmonic potential. Some properties (entropy, temperature, specific heat) of the nanocrystal are calculated numerically but exactly within the framework of the microcanonical ensemble. We find that the presence of a surface in the nanocrystal modifies the thermostatistic properties to a greater extent than the small number of atoms in the system. The specific heat $C_v$ behaves similarly to the Einstein solid, with an asymptotic value for high temperatures that differs from that of the Dulong-Petit law by a term of the order of $N_A^{-1/3}$ and that can be explained easily in terms of the surface. The entropy is non-additive, but this is due to the presence of the surface and we show that the additivity is recovered in the thermodynamic limit. Finally, we find that, when calculations follow the canonical ensemble, results differ little for small systems ($N_A = 27$) and are inexistent for larger systems ($N_A = 1000$).