Density Oscillations in Nonuniform Systems

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The dynamical properties of an inhomogeneous electron gas is a subject with a history almost as long as quantum mechanics. Because the subject is not widely known it will serve as a suitable introduction to this chapter to remind the reader about a few of the major steps. An important early problem was the theoretical understanding of the stopping of a fast charged particle in matter. A charged particle excites the medium with excitation energies covering a wide spectrum from far ultraviolet to soft X-ray frequencies. The stopping power itself does not depend on the details of the spectrum but only on an average excitation energy. The idea came up that one might replace the full dynamical theory by a simplified picture in which the medium was considered as an inhomogeneous electron gas. The charged particle would excite oscillations in the electron gas around its ground-state density and the particle would lose energy by exciting the various modes of excitation in the nonuniform electron gas. These ideas were developed in a classical paper by Bloch(1) in 1933. He developed a dynamical extension of the Thomas-Fermi theory treated in Chapter 1, considering the hydrodynamical oscillations of the density around the Thomas-Fermi ground-state density. Applications by Jensen(2) to a simplified model treating the atom as a small metallic sphere agreed with stopping power data in its dependence on atomic number and supported the hydrodynamical model. The Bloch equations were actually not fully solved until after the Second World War. Many extensions to include, e.g., exchange and correlations have been made. After the development of the density functional method presented in Chapter 2, a hydrodynamical approach based on the density-functional scheme was proposed by Ying et al.(3)