Retardation Corrections Research Articles

Retarded van der Waals Interactions. I. Simple Atoms

The effect of retardation on the dipole dispersion interaction between atoms has been determined. By the use of accurate frequency-dependent polarizabilities, quantitative calculations of the retardation correction to the interaction of (H, H), (H, He+), (He+, He+), (H, He), and (He, He), as well as to the three-body system (He, He, He), have been made. The procedure used was to evaluate the Casimir–Polder integral for two atoms and its three-body extension directly so as to obtain results correct for all R. Although the corrections to the limiting behavior (1 / R6 for “small” R, 1 / R7 for large R) are not large, they may be of significance for high accuracy measurements in some cases.

Retarded Interactions in Fermi Systems

The nature of the two-body interactions in many-fermion systems is studied from the viewpoint of meson theory. An exactly soluble model is formulated for the linear coupling of a meson field to fermion density fluctuations, in which meson degrees of freedom are treated exactly, and fermion motion is treated within the domain of the random-phase approximation (RPA). Instability conditions for the RPA ground state are established. More generally, the effective two-body interaction is deduced via a Green's-function technique by eliminating the meson degrees of freedom. This interaction is shown to be frequency-dependent, i.e., retarded in time. The resulting interaction is then applied to the calculation of the Hartree-Fock (H.F.) field and of the collective modes of the system via a generalized Landau equation. In the H.F. approximation, one obtains an unambiguous separation of renormalization (self-energy) effects and the nucleon-nucleon interactions themselves, the former reducing to the correct mass renormalization of the nucleon in the static limit. For reasonably small momenta ($p<{p}_{F}$), the retardation corrections to the H.F. field can be characterized by a small parameter ${(\frac{{\ensuremath{\epsilon}}_{F}}{\ensuremath{\mu}})}^{2}$ ($\ensuremath{\approx}0.1$ for actual nuclear densities), where ${\ensuremath{\epsilon}}_{F}=\mathrm{Fermi}\mathrm{energy}$ and $\ensuremath{\mu}=\mathrm{meson}\mathrm{mass}$. The corrections become more important at high momenta and densities. In the long-wavelength limit, the frequency-dependent corrections to the collective mode energies are found to be of order ${(\frac{\ensuremath{\omega}}{\ensuremath{\mu}})}^{2}$, where $\ensuremath{\omega}=\mathrm{collective}\mathrm{mode}\mathrm{energy}$. For a static Yukawa interaction, a value ${\ensuremath{\lambda}}^{2}\ensuremath{\approx}5$ (consistent with the usual shell-model values) is found for the neutral scalar coupling constant by requiring that the giant dipole collective state appear at the experimental energy. For pseudoscalar coupling, the usual renormalized coupling constant $\frac{{f}^{2}}{4\ensuremath{\pi}}\ensuremath{\approx}0.08$ is shown to yield a "breathing mode" in heavy nuclei consistent with crude estimates based on nuclear compressibilities.

ELECTRONIC STATES OF MIXED CRYSTALS

Two problems are discussed. On inclusion of electromagnetic retardation corrections in the intermolecular potential energy between impurity and host molecules in a deep-trap mixed crystal a new term appears depending on the inverse square of the distance of separation. Calculations are reported to show that the corresponding divergence in the coupling energy is extremely slow and unimportant in crystals of up to a few centimetres in size. Changes produced by deep-trap impurities in the host absorption spectrum are discussed. The changes in absorption width and Davydov splitting (so far as they depend on rigid lattice effects) are shown to be related to the distribution of levels in the pure crystal exciton spectrum, and are least for energy regions where the levels are most densely packed.

Van Der Waals Attraction between Colloidal Particles: The Retardation Correction for Flat Plates of Arbitrary Thickness

An analysis is given of the van der Waals attraction between two flat plates of infinite extent but of finite thickness, taking account of the retardation correction. Four cases are recognized and the four distinct equations are shown to be mutually consistent. A table of correction factors is given for various values of the particle thickness and separation.