David Pines

First Page

The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.

The behavior of the electrons in a dense electron gas is analyzed in terms of their density fluctuations. These density fluctuations may be split into two components. One component is associated with the organized oscillation of the system as a whole, the so-called "plasma" oscillation. The other is associated with the random thermal motion of the individual electrons and shows no collective behavior. It represents a collection of individual electrons surrounded by comoving clouds of charge which screen the electron fields within a distance of the order of magnitude of the Debye length. This split up of the density fluctuations corresponds to an effective separation of the Coulomb interaction into long-range and short-range parts; the separation occurs at roughly the Debye length.The relation between the individual and collective aspects of the electron gas is discussed in detail, and a general physical picture of the behavior of the system is given. It is shown that for phenomena involving distances greater than the Debye length, the system behaves collectively; for distances shorter than this length, it may be treated as a collection of approximately free individual particles, whose interactions may be described in terms of two-body collisions.This approach is used to study the interaction of a specified electron with the remainder of the electron gas. It is shown that the collective part of the response of this remainder to the field of the specified particle screens this field within a distance of the order of the Debye length; this furnishes a detailed description of the screening process. Moreover, if the specified particle moves with greater than the mean thermal speed, it excites collective oscillations in the form of a wake trailing the particle. The frequency of these collective oscillations and the energy emitted by the particle are calculated. A correspondence theoretical method is used to treat this phenomenon for the electrons in a metal. The results are in good agreement with the experiments of Ruthemann and Lang on the energy loss of kilovolt electrons in this metallic films.The generalization of these methods to an arbitrary interparticle force is carried out, and a criterion is obtained for the validity of a collective description of the particle interactions. It is shown that strong forces and high particle density favor collective behavior, while high random thermal velocities oppose it.

This text continues to fill the need to communicate the present view of a solid as a system of interacting particles which, under suitable circumstances, behaves like a collection of nearly independent elementary excitations. In addition to introducing basic concepts, the author frequently refers to experimental data. Usually, both the basic theory and the applications discussed deal with the behavior of '`'simple' metals, rather than the '`'complicated' metals, such as the transition metals and the rare earths. Problems have been included for most of the chapters.

A variational technique is developed to investigate the low-lying energy levels of a conduction electron in a polar crystal. Because of the strong interaction between the electron and the longitudinal optical mode of the lattice vibrations, perturbation-theoretic methods are inapplicable. Our variational technique, which is closely related to the "intermediate coupling" method introduced by Tomonaga, is equivalent to a simple canonical transformation. The use of this transformation enables us to obtain the wave functions and energy levels quite simply. Because the recoil of the electron introduces a correlation between the emission of successive virtual phonons by the electron, our approximation, in which this correlation is neglected, breaks down for very strong electron-phonon coupling. The validity of our approximation is investigated and corrections are found to be small for coupling strengths occurring in typical polar crystals.