Mathematical Modelling of Radiative Heat Exchanges in Czochralski Crystal Pulling

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Abstract
Crowley has published a mathematical model of the heat flow in Czochralski crystal pulling, assuming the surfaces of solid crystal and liquid meniscus to be cooled by radiation to the atmosphere. An enhanced model is described that takes account of the radiative heat exchanges between the atmosphere and the surfaces of the crystal, the meniscus and the hot crucible wall. For computational purposes, the modelling of the radiative heat exchanges must be sufficiently simplified to allow frequent re-calculation of the relevant heat fluxes in the course of a computer simulation of a Czochralski pulling experiment. Radiation optics suggests a number of simplifications, which are explained. The geometry of the physical system, however, is not one for which standard results exist, and the techniques used to overcome the consequent problems may be of interest in related contexts. The new model is compared numerically with the Crowley model for the case of germanium pulling; its sensitivity to variation of the height of the crucible wall, and its responses to variation of some other parameters are also illustrated.

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