"Madelung model" prediction for dependence of lattice parameter on nanocrystal size

TL;DR

This paper presents a Madelung model predicting how the lattice parameter of ionic nanocrystals like CeO2 and BaTiO3 varies with particle size, aligning well with experimental data.

Contribution

It introduces a simple Madelung model to predict lattice parameter dependence on nanocrystal size, accounting for Coulomb interactions.

Findings

Lattice parameter increases as nanocrystal size decreases.

Model predictions match experimental results for CeO2 and BaTiO3.

Dependence follows a 1/d trend with specific alpha values.

Abstract

The competition between the long range Coulomb attractive and the short range repulsive interaction in ionic nanocrystals creates an effective negative pressure, which causes the lattice parameter a to increase with decreasing nanoparticle size d. A simple ``Madelung model'' is used to predict the dependence of the lattice parameter for CeO2 and BaTiO3 on d, delta a/a = alpha/d, for alpha = 0.22 A and 0.18 A respectively. The model predictions are compared with experimental results.