We deposit phosphorus-doped, amorphous Si by low pressure chemical vapor deposition and subsequently crystallize the films by furnace annealing at a temperature of 600 degrees C. Optical in-situ monitoring allows one to control the crystallization process. Phosphorus doping leads to faster crystallization and a grain size enhancement with a maximum grain size of 15 mu m. Using transmission electron microscopy we find a log-normal grain size distribution in our films. We demonstrate that this distribution not only arises from solid phase crystallization of amorphous Si but also from other crystallization processes based on random nucleation and growth The lognormal grain size distribution seems to be a general feature of polycrystalline semiconductors.