TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Process: The process further involves mixing the feedstock and the impurity-removing agent by bubbling an inert gas, stirring mechanically, and/or agitating electromagnetically. The separating involves density-based liquid separation. The process further involves adding a slag-forming agent. The process further involves reducing a content of the impurity-removing agent in the feedstock. The process further involves solidifying the feedstock during or after separating the high-temperature solid; remelting the feedstock; optionally reducing a content of the impurity-removing agent in the feedstock; and resolidifying the feedstock using directional solidification and separating a last-to-freeze material from a remaining portion. The separating the high-temperature solid from the feedstock involves sinking the high-temperature solid with top-down solidification of the feedstock, or involves trapping the high-temperature solid at a bottom of an ingot with bottom-up solidification of the feedstock. The process involves melting silicon with phosphorus (10 parts per million atomic (ppma)); adding gallium (1000 ppma) to the silicon; reacting the phosphorus with the gallium to form gallium phosphide; separating the gallium phosphide from at least a portion of the silicon; solidifying the silicon; remelting the silicon; reducing a content of the gallium in the feedstock; and resolidifying the silicon. Preferred Components: The high-temperature solid and the molten feedstock have a density ratio of at least 1:1 to more than 1.5:1. The impurity comprises boron or phosphorus. The impurity comprises a concentration of 1 part per million atomic (ppma) and 500 ppma. The impurity-removing agent comprises gallium. The impurity-removing agent comprises 10 ppma and 10000 ppma of a mixture of the feedstock and the impurity-removing agent. The impurity-removing agent comprises a mixture (1000 ppma) of the feedstock and the impurity-removing agent. The slag-forming agent sinks or floats in the feedstock. The slag-forming agent comprises silicon dioxide. The ratio of the impurity-removing agent to the impurity comprises at least 1:1-50:1. The high-temperature solid comprise gallium phosphide. The high-temperature solid comprises a melting point of at least 50 degrees C greater than a melting point of the feedstock. The ratio of the impurities in the feedstock to impurities in a purified feedstock comprises at least 3:1. The mass of purified feedstock comprises at least 75 percent of a mass of the feedstock with impurities. The ratio of segregation coefficients of the impurity to the impurity-removing agent comprises at least 20:1. The impurity-removing agent comprises a segregation coefficient of less than 0.05. The feedstock comprises at least 500 kilograms. The concentration of impurities comprises less than 3 ppma. The high-temperature solid comprises a segregation coefficient of at least 2 or 100 times more effective than the impurity. EXAMPLE - A silicon feedstock (100 kilogram mass) with phosphorus (10 parts per million atomic (ppma)) melts in a crucible. Gallium (1000 ppma) was added as an impurity-removing agent, giving a 100:1 ratio to the phosphorus. The gallium was 99.9 percent pure on an atomic or a molar basis. The gallium reacts with the phosphorus and forms gallium phosphide, a high-temperature solid. During the reaction phase, the melt was stirred to enhance mixing and precipitation. After stirring for 10-30 minutes, the stirring was stopped and the melt was allowed to settle. The gallium phosphide readily separates to the bottom based on phase separation and/or density differences. A silicon rod was then brought into contact with the liquid surface, and an ingot was pulled out of the melt. At the end, 5 percent (based on weight) of the liquid was left in the crucible while the ingot was pulled out of the melt. At this point, the solid silicon has phosphorus (less than 1 ppma) and gallium (30 ppma). The silicon was either sold or melted and directionally solidified one additional time with a greater than 95 percent (based on weight) yield, giving silicon with phosphorus (less than 0.75 ppma) and gallium (less than 2 ppma). The resulting ingot has a usable mass of 90 kilograms (90 percent based on weights) of the original mass of the silicon and the remaining 10 kilograms was scrap.