Excimer laser crystallized Si thin films on fused silica substrates exhibit a peak in the average grain size as a function of laser energy density. The average grain size increases with increasing laser fluence until a maximum value , approximately 10 microns for a 100 nm thick Si film, is achieved. The peak in grain size is accompanied by a peak in the electron Hall mobility. Further increases in the laser fluence result in a decrease in the Si grain size and an increase in the intragranular defects. A small energy range of 40 mJ/cm2 exists in which this peak in grain size can be achieved. Cross section TEM has shown that when the peak laser fluence is exceeded, the fused silica substrate can be as rough as 17 nm. Atomic force microscopy. performed on the substrate surface after the Si has been etched off, also shows that the magnitude and spatial frequency of the roughness increases when the critical laser fluence is exceeded. This degradation of the interface may also produce sites for stacking faults to form during the solidification of the Si. This result and results of simulations of the temperature of the interface during crystallization suggests that the peak energy range exists after the complete melting of the Si thin film and before the silica substrate starts to soften.