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Substantial enhancement of important properties of materials is manifested when impurity concentrations are reduced to extremely low levels. Current technologies are not fully adequate and the frontiers of our present knowledge of theory and practice need exploration. New concepts and methodologies are evolving and novel purification processes are beinb developed for producing ultra-high-purity metals with purpose-designed atomic patterns for sophisticated functional applications. This book gives a critical, up-to-date evaluation starting with an extended introductory treatise on the fundamentals, followed by a detailed description of the new methods of purification of transition metals and rare earth metals, including their characterization. The subject is approached both from the basic science and applied engineering points of view.