Range of Excited Electrons in Metals

The energy of interaction of a single excited electron with the sea of conduction electrons in a metal has been calculated by a self-energy approach. The imaginary part of the self-energy of the excited electron can be interpreted in terms of a total rate of real collisions with the sea of conduction electrons. By weighting the differential scattering rate by the amount of energy lost in each scattering event, one obtains an expression for the instantaneous rate of energy loss of the excited electron as a function of its initial energy. The extremely strong dependence of this rate on the initial energy is the main result of this paper. For slow electrons, by which we mean those of initial energy smaller than the sum of the Fermi energy and the plasma energy of the electron gas, the rate of energy loss is determined by the small imaginary part of the dielectric constant. For electrons close to the Fermi surface this rate is proportional to ${(p-p_0)}^3$, where $p$ is the momentum of the excited electron and $p_0$ the Fermi momentum; therefore in this range the rate of energy loss is very sensitive to the initial energy. For fast electrons, a new contribution to the rate of energy loss arises due to a pole of the inverse dielectric constant. This new process corresponds to the excitation of plasma oscillations by the excited electron, and causes fast electrons to lose energy very rapidly.