How Good Is the Debye Model for Nanocrystals?

Enrique N Miranda,Geraudys Mora-Barzaga

doi:10.4236/jmp.2019.106042

Abstract

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.

Highlights

The boom of physics at a nanoscopic scale justifies questioning the validity of some of the usual claims when working with systems made up of a few hundred or a few thousand particles
The simplest possible nanocrystal is considered: a basic cubic structure made of atoms that interact through a harmonic potential
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

Summary

Introduction

The boom of physics at a nanoscopic scale justifies questioning the validity of some of the usual claims when working with systems made up of a few hundred or a few thousand particles. The atoms are assumed to be configured in a simple cubic lattice, where each can oscillate in three different directions and with the same frequency in all cases Such a straightforward model served to reveal the importance of the surface for the specific heat of the nanocrystal. By comparing both models with the exact results, it will become clear that they are not suited for systems with thousands of atoms.

A atoms and the total volume is given by

The Debye Approximation

Conclusions