The Einstein nanocrystal

TL;DR

This paper models nanocrystals as small cubic lattices with quantum harmonic bonds, analyzing their thermodynamic properties and surface effects through numerical microcanonical calculations, revealing surface influence and ensemble differences.

Contribution

It introduces a simple cubic lattice model for nanocrystals and analyzes surface effects on thermodynamic properties using microcanonical ensemble calculations.

Findings

Surface significantly affects thermodynamic properties.

Specific heat approaches Einstein solid behavior at high temperatures.

Microcanonical and canonical results differ minimally for small systems.

Abstract

We study the simplest possible model of nanocrystal consisting in a simple cubic lattice with a small number of atoms (NA ~ 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 Cv 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 NA^(-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 (NA = 27) and are inexistent for larger systems (NA = 1000).