On the Origin of Logarithmic-Normal Distributions: An Analytical Derivation, and its Application to Nucleation and Growth Processes

TL;DR

This paper derives an analytical expression for the lognormal distribution from nucleation and growth processes, explaining its origin and applying it to crystallized silicon films.

Contribution

It provides the first analytical derivation of the lognormal distribution from basic nucleation and growth principles.

Findings

Derived an analytical lognormal-like distribution from nucleation and growth.

Applied the model to silicon film crystallization data.

Validated the distribution with experimental grain size data.

Abstract

The logarithmic-normal (lognormal) distribution is one of the most frequently observed distributions in nature and describes a large number of physical, biological and even sociological phenomena. The origin of this distribution is therefore of broad interest but a general derivation from basic principles is still lacking. Using random nucleation and growth to describe crystallization processes we derive the time development of grain size distributions. Our derivation provides, for the first time, an analytical expression of the size distribution in the form of a lognormal type distribution. We apply our results to the grain size distribution of solid phase crystallized Si-films.