Characteristic Energy Loss of Electrons Passing through Metal Foils. II. Dispersion Relation and Short Wavelength Cutoff for Plasma Oscillations

Previous work is extended so as to provide a determination of the shortest wavelength at which plasma oscillations can be sustained by a degenerate electron gas. The collective oscillations are treated without introducing collective coordinates, thereby avoiding possible complications associated with subsidiary conditions. Instead the Hartree self-consistent field method is used to provide a direct quantum-mechanical analog to the Bohm-Gross derivation of the dispersion relation. The effect of electron exchange, calculated by replacing the Hartree by the Hartree-Fock method, somewhat decreases the dependence of the plasma frequency on wave number. The maximum wave number corresponds to the momentum just sufficient to cause an electron at the surface of the Fermi sea to make a real transition, absorbing one plasma quantum of energy. This criterion agrees well with Watanabe's measurement of the maximum angle by which electrons undergoing the characteristic energy loss are scattered. The previous work on the intensity of the characteristic energy loss as a function of angle is supplemented by a study near cutoff, where is is shown that the intensity drops rapidly to zero as the maximum angle of scattering is approached. By use of the generalized sum rule of Nozi\`eres and Pines it is also possible to study the contribution of one-electron excitation to the differential stopping power. Varying as the fourth power of the angle of scattering, this contribution is negligible at small angles and first becomes dominant as the cutoff is approached.