MATHEMATICAL EQUATION FOR IMPURITY DISTRIBUTION AFTER SECOND PASS OF ZONE REFINING

A mathematical equation has been derived that describes impurity distribution in ingot after second pass of zone refining. While an exponential impurity distribution is calculated by a simplified model after first pass, second pass is described by mixed linear - exponential model. Relationship of transformed impurity concentration is constant over whole length of semi-infinite ingot for first pass. However, it has linear trend for second pass. Last part of molten zone at infinity solidifies differently and can be described mathematically as directional crystallization. A mathematical tool devised for second pass of zone refining can be tried to be used for derivation of functions of more complex models that would describe impurity distribution in more realistic way compared to simplified approach. Such models could include non-constant distribution coefficient and/or shrinking or widening molten zone over a length of ingot.