Editing Zone melting (section)

===Mathematical expression of impurity concentration===
When the liquid zone moves by a distance <math>dx</math>, the number of impurities in the liquid change.  Impurities are incorporated in the melting liquid and freezing solid.<ref>[[James D. Plummer]], Michael D. Deal, and Peter B. Griffin (2000) ''Silicon VLSI Technology'', Prentice Hall, page 129</ref>{{Clarify|reason=Provide a geometric interpretation|date=May 2022}}

:<math>k_O</math>: segregation coefficient
:<math>L</math>: zone length
:<math>C_O</math>: initial uniform impurity concentration of the solidified rod
:<math>C_L</math>: concentration of impurities in the liquid melt per length
:<math>I</math>: number of impurities in the liquid
:<math>I_O</math>: number of impurities in zone when first formed at bottom
:<math>C_S</math>: concentration of impurities in the solid rod

The number of impurities in the liquid changes in accordance with the expression below during the movement <math>dx</math> of the molten zone
:<math>dI = (C_O - k_O C_L) \, dx\;</math>
:<math>C_L = I/L\;</math>
:<math>\int_0^x dx = \int_{I_O}^I \frac{dI}{C_O - \frac{k_O I}{L}}</math>
:<math>I_O = C_O L\;</math>
:<math>C_S = k_O I / L\;</math>
:<math>C_S (x) = C_O \left ( 1 - (1 - k_O) e^{- \frac{k_O x}{L} } \right )</math>