Static electric fields in vacuum

This chapter begins using Coulomb’s law to derive Maxwell’s electrostatic equations for a vacuum. In doing this, the integral forms of the electrostatic potential Ф and field E are obtained. These results are then used to determine Ф and E for various charge distributions possessing some symmetry: either via Gauss’s law or by directly integrating a known charge density over a line, surface or volume. Applications which require the use of computer algebra software (Mathematica) are included. A multipole expansion of the potential Ф leads to the various multipolemoments of a static charge distribution. Examples which deal with important properties like origin independence are presented. A range of questions and their solutions, not usually encountered in standard textbooks, appear in this chapter.

Relaxations of molecular charge distributions and the vibrational force constants in diatomic hydrides

The force constants for LiH, HF, NaH, and HCl are calculated from Hartree–Fock wavefunctions by a polynomial fit of the forces exerted on the nuclei as a function of the internuclear separation. The magnitude of the force constant is interpreted in terms of the relaxation of the molecular charge distribution induced by the nuclear displacement. In LiH or NaH, for which the molecular charge distribution exhibits the characteristics of ionic binding, two distinct relaxations are evident: a relaxation in the region of the cationic core and a relaxation of the density localized on the proton. The relaxation of the charge density in the vicinity of the Li+ or Na+ core opposes the motion of either nucleus while the relaxation of the density localized on the proton facilitates the displacement of the nuclei. In HF or HCl the relaxation of the molecular charge distribution is dominated by one continuous region of charge increase (for bond contraction) or decrease (for bond extension) over the whole of the binding region, a relaxation which facilitates the motion of the nuclei. Thus the relaxation of a molecular charge distribution and its effect in determining the magnitude of the force constant is dominated by the same features of the static charge distribution which serve to distinguish ionic from covalent binding.

Theory of light absorption and non-radiative transitions in F -centres

A quantitative theory for the shapes of the absorption bands of F -centres is given on the basis of the Franck-Condon principle. Underlying the treatment are two simplifying assumptions: namely, ( a ) that the lattice can be approximately treated as a dielectric continuum; ( b ) that in obtaining the vibrational wave functions for the lattice, the effect of the F -centre can be considered as that of a static charge distribution. Under these assumptions, it is shown that the absorption constant as a function of frequency and temperature can be expressed in terms of the Bessel functions with imaginary arguments. The theoretical curves for the absorption constant compare very favourably with the experimental curves for all temperatures. Also considered in the paper are the probabilities of non-radiative transitions, which are important in connexion with the photo-conductivity observed following light absorption by F -centres. The treatment given differs from the qualitative considerations hitherto in one important aspect, namely, the strength of the coupling between the electron and the lattice is taken into account. The adiabatic wave functions for the F -centre electron required for the discussion are obtained by perturbation methods. The probability for an excited F -centre to return to its ground state by non-radiative transitions is shown to be negligible; similar transitions to the conduction band are, however, important if the excited state is separated from the conduction band by not much more than 0·1 eV. The temperature dependence of such transitions is complicated, but, for a wide range of temperatures, resembles e - w/k T . Tentative estimates show that the result is Consistent with the observed steep drop of the photo-conductive current with temperature.