Zur Berechnung von Spektralfunktionen geeignete Lösungsmethoden der Boltzmann-Gleichung

Abstract

The shape of well-resolved spectral lines for emission, absorption, and scattering phenomena in gases is studied. In section 1 the problem of calculating spectral functions within the framework of kinetic theory is formulated after some remarks on quantum mechanical operators and their nonequilibrium averages have first been made. Section 2 is devoted to a discussion of the relevant generalized Boltzmann equation (kinetic equation) which has to be solved in order to obtain the spectral function. In section 3, the case of high and medium densities is treated by application of the moment method. The resulting spectral function is given by a Lorentzian. Its half-width, in general, contains contributions directly as well as inversely proportional to the number density; the former accounts for collisional broadening, the latter for diffusional broadening. Section 4.1 deals with the opposite limiting case of low densities where the line shape is determined by a Doppler broadened Gaussian. In section 4.2 a criterium is given for the occurrence of a minimal line width (narrower than the Doppler width) at an intermediate density (Dicke effect). Section 5 is concerned with the calculation of spectral functions valid for all densities. In section 5.1 a model collision term with two relaxation frequencies is introduced. Some special choices made previously for these two model parameters are mentioned. Then, in section 5.2, a kind of a variational procedure is outlined which yields a model collision term which is a best approximation to the exact collision term. Furthermore, the relaxation frequencies—one of which depends on the frequency and the wavevector—are related to the exact collision term. In section 5.3, the desired spectral function is calculated from the kinetic equation with the approximate collision term. Finally, in section 6, it is indicated how “best” model collision terms with more than two relacation frequencies can be obtained.