Relaxation Effects in Spectra: Eigenvalue Treatment of Superoperators

Superoperators have been used in several recent papers to treat the effects of relaxation on the shape of spectrum lines. Although the emphasis in these papers has been on M\"ossbauer spectra, the generality of the treatments allows them to be applied in many fields of spectroscopy. To apply these theories it has been necessary in most cases of interest to plot out the spectrum, point by point, inverting a relatively large matrix at each point. In this paper it is shown that parameters of the spectrum, such as linewidths, may be obtained directly from the eigenvalues of the superoperators used in these theories. With the help of perturbation theory, some general results are then obtained in the limiting cases of slow and fast relaxation rates. This eigenvalue treatment greatly simplifies problems such as determining how a particular term in the Hamiltonian affects the shape of the spectrum.