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Information on the scattering of x-ray and γ-ray photons has profoundly influenced the development of our current concepts concerning the ultimate structure and behavior of matter. The essential duality of waves and particles emerged in an especially direct and clear way from A. H. Compton’s interpretation in 1923 of the shift in wavelength which he and others observed when x-rays are scattered by atomic electrons. X-rays, whose wave properties had accounted for their diffraction by gratings and crystals, were shown to possess also the corpuscular properties of energy and momentum, and to be capable of utilizing their entire energy and momentum in a single collision with one particular electron. This dualism of wave and corpuscular properties was extended to electrons and other conventional particles by de Broglie in 1924, and soon thereafter became a cornerstone in Schrödinger’s wave mechanics, in Heisenberg’s uncertainty principle, and in much of the new physics which was soon built by Bohr, Dirac, and many others. The validity of negative energy states of the electron was shown several years before the free positron was discovered, by the experimental confirmation of the Klein-Nishina cross sections which had been developed from Dirac’s relativistic electron theory.