# 柴可拉斯基法

## 其他杂质的引入

### 杂质引入情况的数学描述

${\displaystyle k_{O}}$ ：偏析系数
${\displaystyle V_{0}}$ ：初始体积
${\displaystyle I_{0}}$ ：杂质的初始数量
${\displaystyle C_{0}}$ ：熔融物中杂质的初始浓度
${\displaystyle V_{L}}$ ：熔融物的体积
${\displaystyle I_{L}}$ ：熔融物中杂质的数量
${\displaystyle C_{L}}$ ：熔融物中杂质的浓度
${\displaystyle V_{S}}$ ：固态晶体的体积
${\displaystyle C_{S}}$ : 固态晶体中杂质的浓度

${\displaystyle dI=-k_{O}C_{L}dV\;}$
${\displaystyle dI=-k_{O}{\frac {I_{L}}{V_{O}-V_{S}}}dV}$
${\displaystyle \int _{I_{O}}^{I_{L}}{\frac {dI}{I_{L}}}=-k_{O}\int _{0}^{V_{S}}{\frac {dV}{V_{O}-V_{S}}}}$
${\displaystyle \ln \left({\frac {I_{L}}{I_{O}}}\right)=\ln \left(1-{\frac {V_{S}}{V_{O}}}\right)^{k_{O}}}$
${\displaystyle I_{L}=I_{O}\left(1-{\frac {V_{S}}{V_{O}}}\right)^{k_{O}}}$
${\displaystyle C_{S}=-{\frac {dI_{L}}{dV_{S}}}}$
${\displaystyle C_{S}=C_{O}k_{O}(1-f)^{k_{o}-1}}$
${\displaystyle f=V_{S}/V_{O}\;}$

## 参考文献

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