Effect of radiation losses in the growth of optical crystals by the Czochralski process

A study on the control of the cross-sectional area of an optical crystal grown from its melt is presented in this paper. The optical crystal, participates in the heat exchange between the crystal and its surroundings. Heat is thus transferred to the enclosure from the entire volume of the solid crystal and its melt. Two parameters that influence the size of the crystal as well as its uniformity are the pull velocity and the base temperature of the crucible. The effect of including radiation from the bulk of the material on these control parameters has been examined in the present study. The mathematical model is based on one dimensional unsteady heat conduction in the solid phase in conjunction with radiative heat transfer to its' ambient, by considering the melt and the grown crystal as participating media. The model can predict the time-wise variation of the pull velocity and the base temperature required to maintain the radius of the crystal a constant to within a prescribed tolerance. The governing equation and the radiative heat transfer equation have been numerically solved, using an implicit finite difference method. Materials considered for analysis are YAG and Nd:YAG. The analysis can account for variation in transport properties with temperature and differences in properties between the liquid melt and the crystalline solid. Results of the analysis bring out the importance of including radiation from the interior of the crystal, and the role of scattering in the crystal growth process.