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

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.