Transfer of line radiation in a magnetic field

Abstract

Using a classical approach, we present a derivation of the transfer equations for spectral line radiation in a medium which is permeated by a magnetic field. We discuss solutions of these equations for the " weak field " case when the Zeeman splitting is a fraction of the Doppler width and consider the range of validity of such solutions. We discuss and extend some approximate expressions, and develop some new ones, which allow a simple inference of the vector field characteristics directly from the line profiles. Subject headings: line formation-polarization-radiation transfer-Zeeman effect I. INTRODUCTION This paper discusses some aspects of the formulation, solution , and application of the transfer equations for spectral line radiation in a dielectric medium permeated by a magnetic field. The derivation of these equations, which specify the evolution of the Stokes vector for the line, is approached from a classical standpoint. This allows for a derivation wl^ich is more readily understandable than the quantum-mechanical treatment and in particular allows us to appreciate the physical origin of the various terms. As it turns out, the results of the classical theory are almost identical to the more exact treatment, and we need only reinterpret some of the physical variables which enter the classical theory to obtain, precisely, the quantum-mechanical results. The literature on the subject is sprinkled richly with errors-often in the signs of the terms-and abounds in nomenclatural confusion and obscurity, and it may be worth setting down in one readily accessible place a set of equations which is believed to be correct, along with a detailed accounting of how the terms arise so that the hesitant may check for themselves. In keeping with the basic classical approach, the formulation is developed for a Zeeman triplet, but its extension to more complex splitting patterns is quite simple, and the procedure for doing so is indicated, along with such other modifications to the classical results as are needed. The absorption components of the Stokes transfer equations follow at once from a knowledge of the way in which the amplitudes and phases of the polarized components of line radiation are modified in propagation through an element of length of a magnetized gas. In the classical picture, amplitude changes arise from the excitation, by the radiation, of dipoles in the gas-and subsequent decay of the motion of the accelerated charges. Phase changes are associated with this excitation also, and in the presence of a magnetic field these differ from one component to another, which leads to birefringence and to the so-called magneto-optic effects. The emission components arise from acceleration of the dipole charge by a process which