The effect of an axial magnetic field on periodic growth rate fluctuations in Czochralski crystal growth

In this paper we develop a mathematical model to investigate the effect of growth rate fluctuations in magnetic Czochralski crystal growth. The pull rate is assumed constant and the growth rate fluctuates with frequency ω rad/s, the crystal rotation rate. We explore the effect of the magnetic field and the crucible rotation on fluid flow and mass transfer in the vicinity of the crystal/melt interface. Numerical results are obtained for 0≤N≤32 and -0.1≤s≤1.0, where N is the magnetic interaction parameter and s is the ratio of crucible to crystal rotation. Asymptotic results are obtained for the velocity field for large N and agreement is good for N>4. The analysis is extended to the concentration field to give asymptotic results for N→∞, Sc→∞, N/Sc2→∞ where Sc is the Schmidt number. Again the asymptotic results agree well with the numerical solutions. We conclude that the phase of the resulting concentration fluctuations appears to be independent of N and s. The spatially averaged effective distribution coefficient, keff and the amplitude of the concentration oscillations, Ac, are linked. As keff→1, Ac increases and so improved macrosegregation is accompanied by a deterioration in microsegregation