柴可拉斯基法

其他杂质的引入

杂质引入情况的数学描述

$k_O$：偏析系数
$V_0$：初始体积
$I_0$：杂质的初始数量
$C_0$：熔融物中杂质的初始浓度
$V_L$：熔融物的体积
$I_L$：熔融物中杂质的数量
$C_L$：熔融物中杂质的浓度
$V_S$：固态晶体的体积
$C_S$: 固态晶体中杂质的浓度

$dI = -k_O C_L dV\;$
$dI = - k_O \frac{I_L}{V_O - V_S} dV$
$\int_{I_O}^{I_L} \frac{dI}{I_L} = -k_O \int_{0}^{V_S} \frac{dV}{V_O - V_S}$
$\ln \left ( \frac{I_L}{I_O} \right ) = \ln \left ( 1 - \frac{V_S}{V_O} \right )^{k_O}$
$I_L = I_O \left ( 1 - \frac{V_S}{V_O} \right )^{k_O}$
$C_S = - \frac{dI_L}{dV_S}$
$C_S = C_O k_O (1-f)^{k_o - 1}$
$f = V_S / V_O\;$

参考文献

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