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Czochralski method
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===Mathematical form=== Impurity concentration in the final solid is given by <math display=block>\frac{C}{C_0} = k\left(1-\frac{V}{V_0}\right)^{k-1}\text{,}</math> where {{mvar|C}} and {{math|''C''<sub>0</sub>}} are (respectively) the initial and final concentration, {{mvar|V}} and {{math|''V''<sub>0</sub>}} the initial and final volume, and {{mvar|k}} the [[segregation coefficient]] associated with impurities at the melting phase transition. This follows from the fact that <math display=block>dI = -k_O C_L dV</math> impurities are removed from the melt when an infinitesimal volume {{math|d''V''}} freezes.<ref>James D. Plummer, Michael D. Deal, and Peter B. Griffin, ''Silicon VLSI Technology,'' Prentice Hall, 2000, {{ISBN|0-13-085037-3}} pp. 126β27</ref> {{Portal|Manufacturing}} {{Commons category|Czochralski method}}
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