Theory of Collective Oscillation of Electrons in Solids

Abstract

A general formulation of the collective oscillation of electtons in solids are developed to take into account fully effects of ion-core potential; Based on the second quantized scheme of the many-body .system, die collective component of density fluctuation is examined carefully and it is shown that we can define a set of the collective normal c:CHilCiinates under the specified conditions. In Section 3, the many-electton Hamiltonian is transformed into the Hamiltonian of the collective field and of the electtons With a modified electron-electton interaction and the intetaction Hamiltoliian between the collective field and the electtons. In Section 4, we derive the formula of the generalized plasma frequency and that of the dispetsion relation. Recent ·developments of experimental studies of the scattering of fast electrons by thin solid films have made it possible to observe the energy spectrum of scattered electrons with extremely high energy-resolution. The spectrum consists of several lines, of which higher order lines appear to be multiples of a fundamental one. Since the line width is fairly narrow, it seems very hard to interpret the result in terms of single electron excita­ tion through the inter-band transitionsY Thus, Bohm and Pines91 have suggested a possible interpretation in terms of the excitation of a . collective oscillation in solids by the interloper electrons. The fundamental nature of the collective oscillation of electrons in metals has been studied theoretically by Pines.3> As the Bohm-Pines theory dealt with the free electron system, many efforts•>,?> have been forwarded for the extension of the Bohm-Pines theory to take into account the effects of the periodic potential- of atomic cores. The purpose of this paper is to present a general formulation of the collective motion of electrons. in solids, based on the realistic physical picture that the collective motion in solids is due to the density :fluctuation of electrons. So far all of the works on the present subject have used the canonical transformation method, which has restrictive limi­ tation in the application, to formulate the present ·problem. In the canonical transformation method, at first place one has to write down a Hamiltonian that describes the motion of the electrons and an auxiliary plasma field, then one tries .to show this Hamiltonian is formally equivalent to the real Hamiltonian of the .. many-electron. s~ und~ some subsidiary condition. This procedure can be