How Good Is the Debye Model for Nanocrystals?

The question here is whether the Debye model is suited to evaluate the specific heat of nanocrystals. For this, the simplest possible nanocrystal is considered: a basic cubic structure made of atoms that interact through a harmonic potential. This simple model can be solved exactly. This allows the dispersion relation of the mechanical waves to be determined, so that calculating the exact specific heat turns out to be quite straightforward. Then, the same problem is solved using the Debye approximation. Our findings show that the specific heat of a nanocrystal evaluated exactly is higher than the value found in the thermodynamic limit, that is to say, the specific heat decreases as the nanocrystal size increases. In addition, it becomes clear that the Debye model is a poor approximation for calculating the specific heat of a nanocrystal. Naturally, the Einstein model yields an even worse result. The cause of the discrepancy is found in the role of the nanocrystal surface.

Boltzmann or Gibbs Entropy? <br/>Thermostatistics of Two Models with Few Particles

We study the statistical mechanics of small clusters (N ~ 10 - 100) for two-level systems and harmonic oscillators. Both Boltzmann’s and Gibbs’s definitions of entropy are used. The properties of the studied systems are evaluated numerically but exactly;this means that Stirling’s approximation was not used in the calculation and that the discrete nature of energy was taken into account. Results show that, for the two-level system, using Gibbs entropy prevents temperatures from assuming negative values;however, they reach very high values that are not plausible in physical terms. In the case of harmonic oscillators, there are no significant differences when using either definition of entropy. Both systems show that for N = 100 the exact results evaluated with statistical mechanics coincide with those found in the thermodynamic limit. This suggests that thermodynamics can be applied to systems as small as these.