On the generalization of the Appleton-Hartree magnetoionic formulas

Abstract

The complex refractive index and the state of polarization in a weakly ionized gas with an alternating electric field and a steady magnetic field are given by the ordinary Appleton-Hartree formulas. In the original derivation a ‘frictional’ term is utilized which is assumed to be independent of the electron velocity, v, and the electron velocity distribution. It represents a first approximation to an effective collision frequency, vAH, for the collisions between electrons and neutral molecules. The present work is an extension of Jancel and Kahan's magnetoionic theory, which is based upon solutions of the Boltzmann equation, when v = vmf(v). The expression for the complex refractive index and the state of polarization are rederived, utilizing a generalized conductivity tensor for the Lorentz gas. The resulting solutions are shown to be identical with the ordinary Appleton-Hartree formulas when v = constant. In the general case, v = vmf(v), a new angular dependent term appears, the coefficient of which vanishes, when v = constant. The elements of the generalized conductivity tensor are integrals involving the electron velocity distribution function. The general non-Maxwellian distribution function for the electrons is derived as a function of the alternating electric field and a steady magnetic field, when the two field vectors have an arbitrary inclination to each other. In the ionospheric wave propagation, the electrons are assumed to have a Maxwellian velocity distribution, as the electric and magnetic field effects will be negligible. The elements of the generalized conductivity tensor are then expressible in terms of previously tabulated integrals, when use is made of Phelps and Pack's laboratory results, viz., v ∝ v2 in air. This greatly eases the computational use of the generalized formulas. Calculations have been carried out for longitudinal and transverse propagation in the cases, vAH=vm=110ω,vAH=vm=ω/2, and vAH = vm = 2ω, and with s (electron gyrofrequency) the same order of magnitude as ω. Generally the birefringent properties of the medium are decreased, when the velocity dependence of the collision frequency is taken into account through the general theory. In all cases the absorption factors based on the generalized theory differ from those based on the ordinary Appleton-Hartree formulas by amounts from 30 to 100 per cent. Improved agreement is obtained when vAH in the Appleton-Hartree formula is associated with the mean energy instead of the most probable energy as in the generalized theory; i.e. when vAH=32vm instead of vAH = vm. In the asymptotic limit, v ≪ ω ± s, the ordinary Appleton-Hartree formula can be retained, provided that vAH=53×32vm=52vm. In the other asymptotic limit, v ω ± s, these same formulas can also be retained when vAH=32vm. For the intermediate case, v ∼ ω ∼ s, differences in the absorption factors between the two theories persist with amounts up to 100 per cent, even though vAH=32vm. It is concluded that the generalized theory should be utilized in this case for all precise experimental work.