Analysis of the transition from the Einstein crystal model to the Debye model

We discuss the low-temperature specific heat of a system consisting of a small number of particles constituting a simple lattice and analyse its dependence on the number of particles, on their frequency distribution and on the dimensionality of the crystal. The results are compared with those expected either in a non-interacting particle model (Einstein crystal) or in a continuous frequency distribution system (Debye model); they show that a number of particles of the order of a few tens is sufficient to give a good agreement with the Debye model. The analysis of the effects of the frequency split indicates that the Debye model is satisfied when the frequency separation is of the order of 80% of the fundamental frequency. The dependence of the results on the sample dimensionality is discussed. The results are valid for other thermodynamic functions such as the energy or the entropy.