Foundations for a Treatment of the Scattering of Light by the Hydrodynamical and Statistical Atom Model

Abstract

A study has been made on the scattering of light by the hydrodynamical and statistical atom model. Bloch's treatment of the hydrodynamical equations of motion for this model is supplemented here by inclusion of the interaction with the electromagnetic field. We limited attention to oscillations of small amplitude. By correspondence principle arguments, general expressions were derived for the cross sections for absorption, coherent and incoherent scattering. The energy can be expressed—following Bloch—as the energy of a Thomas-Fermi atom plus a Hamiltonian which is associated with departures from the Thomas-Fermi distribution. Using Bloch's quantization of this Hamiltonian and applying the method of quantum field theory, we rederived the correspondence principle results for elementary cross sections. Then applying the correspondence theoretical argument to the matrix element, we rederived Heisenberg's result for the total intensity of the Compton scattering. We also apply the method of stationary phase to the hydrodynamical treatment and show that this method gives the same result as does the linear term in the momentum transfer in Heisenberg's expression, except for a numerical factor 3½/2—a point that was discussed by Bloch many years ago. Application of the general formulas given here for angular distribution of Rayleigh and Compton scattering will require electronic machine calculations of higher modes of oscillation of the gas model of the atom analogous to those made by Wheeler and Fireman for l=1.